"I am thinking about a Rimfire rifle in a cartridge other than .22lr. "
So, how about a lever action magnum rimfire?
I used to have a Marlin model 39 in 22 Long Rifle, and a buddy of mine had a SEARS lever action rifle that I believe was made by Marlin and identical to their model 39 in 22 Magnum , and I test fired that with my friend. We were both very impressed with it.
Winchester and Henry both have lever action rimfire rifles, too. Are some offered in 22 Magnum or 17 Hornady magnum chamberings ?
There's a very nice lightly used Winchester 190 in 22 Magnum on the rack at Country Folks Superstore in Cumming right now. I think the price tag on it is $180.