If state police try to arrest federal agents for making what they believe are unlawful arrests, federal agents presumably will respond by arresting the state police for what they believe are unlawful arrests of them. Do governments really want to go there?

I should add that what normally prevents this from being a problem is that there are deference doctrines both ways. If the states want to prosecute feds, the case gets removed to federal court and Supremacy Clause immunity applies, which seems to be a lot like qualified immunity.

Likewise, when the feds want to prosecute the states, they can bring federal criminal actions for constitutional violations by state officers, but it's subject to a qualified immunity-like standard. See United States v. Lanier, 520 U.S. 259 (1997).

If I'm reading the cases correctly, each can prosecute agents of the other, but only for really blatant violations. I gather this ordinarily prevents a tit-for-tat, as prosecutions only happen in extreme cases. But there are aspects of this situation that are not ordinary.