You don't present it "to court" because that might be decided by a jury of unsophisticated people. You file an Inter Partes Review, which goes to the Patent Trial and Appeal Board, and is cheaper (note I didn't say "cheap").
The U.S. doesn't have a "loser pays" model like other countries. You can file for attorney's fees, but the barriers are higher.
I've never heard the term "invalidity counterclaim" and I was in Google Patent Litigation where suits come in constantly. IPRs were pretty standard. Maybe the lawyers did use it and I just never heard it.
If something was going to trial, then "invalidity" was one of the possible defenses, as was "non-infringement."
Who are you making this claim TO? The court? The PTO?
The U.S. doesn't have a "loser pays" model like other countries. You can file for attorney's fees, but the barriers are higher.