Now while there is exceptions to every rule I've found that ammo of similar quality to also be very similar in price. Often post come up about which cartridge for varmints/predators, .223 or .22-250. Invariably someone will toss out that the .223 is much cheaper to shoot than the .22-250. That may be true if you're comparing cheap imported ammo to quality hunting ammo but once you start comparing apples to apples the price get much closer. Not saying to not get a .223, just realize that while deals can be found you're not gonna save big money shooting one compared to many other centerfires. On the ther hand, the .223 is a good choice for many things including punching paper. For the Op's purposes it's a very good choice even if ammo price wasn't a consideration.
As for the magmun rimfires......... OP stated capable of deer? Kinda tosses those outta the picture.
I like the idea of a .38/.357 rifle of some flavor. But while some can be remarkably accurate on average you're not gonna shoot tiny bug holes with one.