I'll revise my earlier post.

A Hornet would be handy, with soft points and TSX's for the smaller varieties - so it would be a wide range of bullets & of course, handloading allowed - only 2 powders needed for all curcumstance and game - Lil'Gun for the Hornet and H4895 for the .375:
.22 CZ Hornet 'Lux" - 40gr. - 45gr. SP's and X - 3,200fps to 3,300fps

.375/06 IMP - 235gr. TSX to 300gr. solids (any make that works) - 2,840fps to 2,450fps

12 bore with variety of loads.