Personally I'd just start a new build. Find a used 10/11 or 110 sporter and cut it down to 16". I'd make it a .308Win. For the puropses you describe.
I did one several years ago and took an old flatback model 11 in .243, bought a synthetic DBM takeoff stock and a takeoff .308 sporter barrel from the classifieds here. I tuned the trigger with a lighter spring, added a home made bolt lift kit cut the stock down to 12" LOP and bedded it reshaped the factory recoil pad cut the barrel down to 16" and had it threaded and braked. Also added a Carstans style comb riser made from kydex with bolts and knobs from the hardware store. Then I gave the whole thing a khaki and olive drab sponge camo paint job with a clear dull coat.
The resulting super handy little carbine is one of my favorites.well when I can pry it away from the grandkids nieces and nephews.
Of course you could always take the economy option. Check the classifieds and find a takeoff sporter in the caliber of your choice and cut it down.
As far as velocity loss, nothing you shoot with it will know the difference to any reasonable hunting range.
Good Luck