He's definitely injured, and it's kinda urgent. If you go to the emergency vet, that will cost 5x.
I don't trust shelters not to put cats down. They really do not GAF.
This is a bad week to need to see a doctor though :-/ at least you've got him contained. Injured cats will hide until they die, so keep him safe inside. He may have difficulty "eliminating" with a broken tail. If he isn't going, it's life threatening. Take him to the ER if so.
Sucks how expensive everything is.