Technically speaking, any union can be dissolved if the people on the other side of the table are willing to cope with the consequences. A union is merely a collective bargaining structure that negotiates employment contracts as a group. The private sector in the US has a long history of union busting, which generally involves firing all union-affiliated employees — ending their contracts peremptorily — and suffering though months of acrimonious strikes and protests. It becomes a matter of endurance and attrition: the business hopes that loss of income will force striking workers to seek employment elsewhere, thinning and eventually dispersing the protests; the union hopes that strikes will limit production by depriving the business of workers, and impact sales by impugning the business' public reputation, and that the consequent loss of profits will force the company to the bargaining table.
In the early days, union fights were far more violent. Businesses would sometimes hire private police to intimidate or rough up individual union members and leaders, or to physically disperse strikes; this was one of the main sources of revenue for the Pinkerton Detective Agency in the late 19th and early 20th centuries. Likewise, union members would sometimes intimidate and rough up 'scabs' — temporary workers hired by the company — or damage equipment (monkey-wrenching) to slow or prevent production. Those tactics (mostly) disappeared in the US with the National Labor Relations Act in 1935, and while many modern corporations still resist union formation (e.g. Amazon, which is aggressively opposing unionization), most companies that already have unions find collective bargaining more palatable than the alternative.
There is nothing intrinsically different about a public-sector union: i.e., unions for people working in essential public services like police, firefighters, air traffic controllers, metro workers, or the like. People employed in these areas face the same challenges and risks of exploitation as people in the private sector, and collective bargaining is a useful tool to ensure they receive adequate compensation and decent workplace environments. And yes, states and municipalities could decide to simply break the collectively-agreed upon contract, firing all workers who do not agree to new terms and suffering through the inevitable strikes and protests. However, there are two factors relevant to public-sector unions that are not seen in the private sector:
The ostensible 'employers' of public-sector workers are not private individuals but publicly elected officials; this gives public-sector unions a purely political impact that increases their bargaining power. A private-sector union is forced to deal with the management as given of the company they work for; a public-sector union can pour its resources into changing the management to something more sympathetic to its interests, using its political clout to elect people who will be inclined to agree with its demands.
The public itself becomes a hostage in union negotiations. Unlike a private-sector union squabble, in which the public might (at worst) be forced to turn to a different supplier for the duration, public-sector union squabbles can become immediate threats to the health and safety of the public at large. The public cannot help but be involved in public-sector union battles, because the public will have to 'do for itself' in the event that union members are fired or walk off the job. And obviously, health and safety are 'unlimited value' items for which almost any demand will be met — this is arguably the same reason that the privatized US health system has such exorbitantly high prices; people will pay what's needed to ensure their health — and this too dramatically increases the bargaining power of a public-sector union.
In short, the problem is not that police unions exists or that they could not be dissolved with a certain amount of risk and effort. The problem is that police unions have interests that are (in some cases) directly opposed to the health and safety of the public they ostensibly serve, and they have a strong bargaining position that allows them (in some cases) to enforce demands that are contrary to the public interest. Forcing police unions out of politics — perhaps as a conflict of interest — would be a good start, but until we make some fundamental legal distinctions between public and private-sector unions, the problem will persist.