There is the concept of malicious prosecution ( https://en.wikipedia.org/wiki/Malicious_prosecution ), but that only applies when the prosecutor pursues a case without proof of even the basic elements of the crime. If they filed charges that they were incredibly unlikely to prove (which is what I heard from all the commenters on this case: “the prosecution has a hard case to make” or something similar), then they spent a lot of time and money, caused somebody to spend a lot of money for the defendant, etc., but they didn’t cross the line into malicious prosecution.
Unfortunately, for this specific case, I don’t think it qualifies. The judge only dismissed the charge because he couldn’t write clear jury instructions for it ( https://apnews.com/article/why-did-judge-drop-kyle-rittenhou... ). I think that could be enough to say “we brought the charge because we honestly thought it applied; the confusion wasn’t malice, it was just difficulty understanding a convoluted law.”
I don’t know about Wisconsin, and I don’t know if Rittenhouse has some way to afford bringing a malicious prosecution lawsuit or some other lawsuit against Kenosha. I get the feeling that kind of reaction (followed by a negotiated settlement) is more common in some parts of the country than others.