A error in reasoning about the pandemic that I see a lot lately.

I don't think this is fallacious. There are too many variables and we are really not talking about causality here; it is more of a political response combined with a claim. For example, the government doesn't have to close schools, make restrictions, etc. in response to a pandemic—not a strict causality sense. To say the pandemic caused the mask mandates, would be inaccurate. Now for the claim, such as the government is to blame for the lockdowns, this has to be supported with argumentation. Defending that argument would almost certainly lead to an insignificant cause fallacy, or one of the related fallacies that deal with claims of causality.

Nirvana fallacy seems to be at play here.  The critic is blaming government because there was not an ideal solution to the Coronavirus pandemic, the choice was between economic hardship and letting people drop dead like flies.  If they had done nothing, the government would also have been criticized, so the government cannot win.

Is there a fallacy specifically for the whole 'cannot win either way' kind of argument?

Rationalization also perhaps, the critic is actually saying that they value the economy over the number of people that would die from an uncontrolled pandemic, which never-the-less isn't going to be high enough to stop the economy functioning.