If an impeachment for something clearly not a ‘high crime or misdemeanor’ were to go though for example for increasing tariffs, would the Supreme Court have the power to overturn their judgement?
Unknown, but unlikely
The constitution uses the word "sole" in only two places:
- The House shall have the sole Power of Impeachment;
- The Senate shall have the sole power to try all impeachments.
There are very few impeachment related judicial precedents, but the decision in Nixon vs. US (this is a Judge Nixon, not the former President) indicated that the use of the word "sole" is exceptionally important, in part because this is the only place it is used in the Constitution, and that the courts really should not get involved in impeachment matters.
Most of the constitution is about building up a system of checks and balances, where a power in one branch is balanced by checks/powers in another. But only in impeachment, which can be applied to both the Executive and Judicial branches, does the constitution seem to single out that there is no corresponding ability for those branches to "check" the power (in large part because this power is itself a check on those branches). Part of the dicta (aka non-binding ramblings) of a concurrence in the aforementioned decision suggested that maybe they could get involved in really extreme cases, namely when the trial doesn't even remotely resemble a trial. While it's not specified, my impression is that this Justice was thinking of an inquisitorial, banana-republic show trial, sort of thing; not a questionable political move or let's-argue-by-dictionary type of problem. The Judiciary has long established that they will presume the other branches are acting in good faith unless there's clear evidence to the contrary, and clear legal power for them to consider it.
Any judicial involvement would immediately run afoul of the Senate's grant of the "sole power to try all impeachments", as now any court getting involved and attempting to change things would be asserting that it, too, has power and jurisdiction over impeachments, and that it can change the Senate's results or order the Senate to change its procedures (and then redo the whole thing). And that opens up a whole can of worms of how Congress is supposed to be able to use impeachment as a check on the Judiciary when the Judiciary can just step in and reverse things. While one can never confidently assert what a hypothetical SCOTUS might rule in a hypothetical situation, frankly your hypothetical doesn't seem to come even remotely close to warranting judicial involvement.
I think the only place where we might expect the courts wouldn't necessarily just bow out and let Congress do whatever it does is on something that grossly violates a constitutional protection. If, for example, the impeachment was specifically and expressly done for the reason "They're a Muslim"—and I'm not talking a hidden reason, but that it is expressly as such on the most superficial and trusting of readings of the impeachment and trial documents—, then now we've got a potential constitutional crisis. The courts probably would want to step in to address a violation of Article VI Clause 3 (the "No Religious Tests" clause) and possibly the first amendment, but yet may not be able to justify themselves as having the power to do so (or even the power to provide relief for the injury). But Congress would have to be really dumb, or weirdly goading the judiciary, to do that.
Note that President Johnson was impeached for violating the 1867 Tenure of Office Act The act explicitly declared that any violation would be a "High Misdemeanor" and was obviously designed to be the ocasion of an impeachment. No court at the time challenged the impeachment as unwarranted. However, in the later case Myers v. United States, 272 U.S. 52 (1926) The US Supreme Court found a similar restriction unconstitutional, and gave its opinion that had the 1867 Tenure of Office Act not been repealed (in 1887) it would have been struck down.
It is hard to argue that violating an unconstitutional law is a "High Crime" or "High Misdemeanor".
I think the conclusion must be that "High Crimes and Misdemeanors" include whatever Congress says that they do, possibly short of a direct and explicit violation of constitutional protections. (If a president were impeached for exercising his constitutional powers, as by giving commands to the military, or for exercising a constitutional right, as for practicing a particular religion, perhaps the Court would intervene.)