In a pure free market nothing should interfere with free trade. Any tax or penalty of any type will be opposed by an advocate of a pure free market , including a carbon tax. So there is no real need to provide an argument against a carbon tax specifically since any tax is counter to the very notion of 'free market'.
However, we don't actually have a free market, or at least not a pure one. We have regulations and taxes on many parts of our market. In truth even most 'free market advocates' usually don't argue for a pure free market, they usually will accept that some small amount of goverment oversight is acceptable, for instance to create a penalty for someone mixing arsenic into their multi vitamins or otherwise doing something blatantly immoral for profits. The real argument is how much goverment oversight can be accepted before it infringes too heavily on free trade.
If we were to hypothesis a free market advocate who accepts some minimal oversight there are a few claims which one could make for arguing why a carbon tax would be too heavy handed:
The Government won't create the tax properly
A key belief driving proponents of free market is the belief that humans in general, and the goverment in particular, are not able to properly anticipate something as complex as the market and thus will not be able to anticipate how their policies they propose will affect the economy. Honestly we have seen plenty of examples of goverment economic policies that failed to achieve the results intended, which suggest that there are limits to the ability to anticipate the affect of policy on the market. Due to the limited ability to fully anticipate repercussions of policies a free market advocate would argue any tax or other economic policy will likely to be poorly adjusted and thus risk doing more harm then good.
Your questions posits a presumption, that a carbon tax is "accurately accounted for", but a free market advocate would counter by saying that the tax wouldn't be "accurately accounted for" because we lack the ability to predict the final result of the tax well enough to be able to determine what an 'accurate accounting' should be. thus one could argue that the tax would risk being too severe and causing a disproportionate harm to businesses, by making it too expensive for a business to make a profit and thus driving businesses to shut down.
Even if the tax is not too severe it may otherwise be poorly created, perhaps by having a poor definition of carbon emissions that disproportionate penalizes certain types of companies, or encourages illogical changes that lower the tax, but not actual emissions. For instance a company may try pushing the emissions into an area that isn't taxed, like buying resources from, or moving manufacturing jobs to, locations outside the USA because the carbon emissions of foreign production are not properly calculated in the taxes, which ultimately lower the companies efficiency but does nothing to address climate change. There are many ways such a tax could theoretically be poorly implemented, and free market advocates generally presume the goverment will not manage to create a good implementation.
This is really the key argument for free market in general, to allow the market to sort things out because humans are so bad at making up good policies, and as such is likely the biggest argument you would here from someone arguing for free market.
The overhead penalty of the tax can be larger then the tax itself
Taxes don't 'just happen'. To tax carbon emissions one needs to know how much carbon is produced. This requires lots of record keeping and logistics. To enforce this law the goverment is going to want to be able to perform audits on companies, to make sure companies don't lie about their emissions, which means a company needs pay people to keep these records, prepare and send them to the goverment, and handle audits if the company is audited.
This all puts a logistical overhead on a company, and it's not trivial! Accounting costs can be a decent chunk of expenditures for some companies, anything adding to the bureaucratic overhead is frowned on by free market advocates.
It can be argued that if the tax is low enough the overhead costs may come up costing almost as much as the tax, and all that bureaucracy is 'wasted' resources that provide no one any benefit are adherent to the spirit of free market (see the Broken Window Fallacy).
Again this isn't limited to carbon tax specifically. There are many potential policies out there that would make sense, and even be accepted by free market advocates, if it wasn't for the added waste of the bureaucracy necessary to implement, and prove the implementation, of the policy which end up being so costly as to counteract the theoretical good of the policy.
The smaller the goverment the better
While not technically required by free market views it's very common for people who argue in favor of free market to also argue for small goverment. The belief is the larger the goverment gets the less efficient the goverment becomes, as bureaucracy, added layers of management, bloat, etc build up.
Therefore adding a bureaucratic overhead on the goverment side, to ensure that carbon taxes are implemented and audit companies, adds direct waste, in man-hours spent proving the policy itself, and adds indirectly to the goverment inefficiency seen as inherent in growing the size of the goverment and it's bureaucracy.
Likewise giving more money to the goverment, in the form of the taxes paid, is not generally seen as a good thing as it's believed the goverment will waste that money while doing little efficiently with it (like paying for the overhead created by the new tax). It's believed the money paid in taxes would do more good if it was kept flowing through the market, encouraging growth based off of basic golden hand of the market, then in the 'wasteful' clutches of an inefficient goverment.
The carbon overhead of bureaucracy can (partially) counteract the carbon savings
All that overhead in implementing the policy will require people traveling to sights, paper printed, electricity powering computers etc which will create carbon emissions. I highly doubt it would be as much as the carbon saved from more efficient policies encouraged by a carbon tax, but it can be argued to lower the potential benefit of the tax, and thus increase the odds that the overhead will do more harm then the benefit of the policy.
Why should the US pay to make the world better for China?
Climate change is a global problem being contributed to by every country in the world, but a carbon tax will only affect the country that enacts it. This could result in US companies suffering under a tax to lower emissions only to have some other country (China being the best bogyman example) to feel free pumping out more carbon counteracting the sacrifice of US companies.
This basically comes down to the classic Tragedy of the Commons style argument. Maybe we would all be better off if we produced less carbon, but no single country wants to make the sacrifice to improve the world if the rest of the world isn't willing to follow suit because no country want's to suffer a disproportionate harm in addressing Climate Change then their fellow countries suffer.
Climate agreements, like the Paris Agreement Trump campaigned on ending (and did), are meant to address this by forcing every country to agree to suffer evenly in addressing the affect of climate change. But since every country in the world is unlikely to sign up to any agreement you will still get this sort of argument (Why should US, Paris, UK, and the other ratifiers of the agreement suffer to fix the Climate when Russia hasn't ratified the agreement but will still benefit from us limiting emissions...). There are also plenty of arguments about rather countries actually adhere to the agreement, to rather such agreements disproportionately harm one country over another, etc etc that play a role.
I shouldn't have to pay for your grandchildren
Ultimately the affects of climate change aren't felt too heavily now, they will be felt in a generation or two once the change has grown enough to have more massive affects. Some can argue that they shouldn't be forced to pay for the well-being of a bunch of people who aren't even born yet out of their own pocket. IE this is a problem for the next generation. Yes you can argue that is a short sighted view, but humans can be short sighted. If you ask someone rather they are okay with loosing their company/job now (as a side effect of taxes slowing businesses and possibly shutting down marginally productive factories) in order to try to make things better in 50 years many would say they would rather be employed now.
I'm cold during winter, obviously global warming must be a myth!
I hate to say it, but climate change denialism is likely to play a huge part in any attempt to pass a carbon tax. I will be the first one to say I've done the research and have no doubt that climate change is real and backed by plenty of scientific evidence; but humans are also pretty good at ignoring science when it tells them something they don't want to hear.
Only 63% of the US believe climate change is a threat, which means 27% believe it isn't one. I disagree with that 27%, but that doesn't matter. To the 27% who don't believe climate change is a threat a carbon tax is 'obviously' an idiotic idea since it creates overhead and weakens companies all while offering no benefit besides making people less afraid of a 'made up bogyman', maybe one created by china to make our companies fail? We can't let China get away with that!
Even when you look at the 67% who believe climate change to be a threat not all believe it's a man made threat, and not all agree that it is a serious threat, only 43% of Americans consider climate change to be a 'very serious' threat. If someone believes the threat minor enough then the minor change created by a carbon tax to the minor harm of climate change may seem too trivial a difference to justify the expense to attempt to optimize.
Studies also show climate change denial is linked with free market beliefs. So a strong proponent of the free market is more likely to either not believe in climate change or believe it to be a minor affect not worth penalizing companies over.