I highly recommend NOT using a rimfire, even though they are legal in some states and even though they were used regularly with the old time hounddoggers. I have seen enough lions killed over my hounds that I tell my friends that come that I strongly encourage a centerfire rifle....to the point that I frankly tell them I will not let my pack follow up a lion they wound with a pistol or bow...they are on their own. I would not let anyone shoot over my dogs with a rimfire unless it is a bobcat, no exceptions there, and I always tie my dogs up for the shot. Yes, a .22 through both lungs is sufficient...or it may not be and none of my dogs' lives are worth gambling whether or not someone hits rib or shoulder blade resulting in a pissed of wounded cat that may decide to fight rather than climb the next run.
The reason I say rifle is that most guys are not good enough with a pistol and a bow is iffy if the cat is high and can't be jumped out for another run to a different, hopefully lower branch, of another tree. A 30-30 is great...slow caliber and makes for an easy shot that will likely make them dead before they hit the ground.
I have seen quite a few of shots flubbed on lions (bow and pistol) and it has formed my thinking on this. What seems like an easy shot is at a weird angle with howling dogs, a pissed off looking lion, a lot of excitement and more often than not, a few people in the audience.
Also, full disclosure, I am super conservative when it comes to the safety of my pack....pretty much a pansy. I figure if I am sending them off after 160 pound cats I can at least make the shot part as safe for them as possible. I also hunt with usually just 2 dogs. Wound a tom lion that falls into 6 or 10 hounds like some guys use and they will kill it and some will need 7 days of antibiotics. If a mature tom fights 1 or 2 dogs it will likely mean 1 or 2 dead dogs.