6mm creedmoor is the answer you are looking for.
With a Sierra Gamechanger or a Barnes LRX, it punches well above it’s recoil class.
Any fast-twist 6 really, the Creedmoor version just makes it easier without going custom. Berger 115s at 3000 (Hodgdon data StaBall 6.5) oughta work for deer, and lighter bullets for vermin.