RE: What Cal. do u think might fit the bill?
The 22 cals. IMO are to small for deer & are varmit rounds.
243's are the starting point for deer sized game with proper bullets. Also doubles as a varmit round.
I use a 25.06 for varmits with 87-90g. bullets & 120g. bullets for deer sized game. IMO one of the better deer rounds.
270's shoot a heavier bullet than the 25.06 at just slower speeds and also is not just a varmit round.
Best Wishes