You can pick the carcass up and drop it by your feet (or move the character on top of the carcass, unless too fatigued). It's possible to do in the current version, although it doesn't do anything about dog stupidity (such as refusing food thrown at them only to bark of hunger a few minutes later).
My typical approach with any game that's small enough to carry with relative ease is to pick them up, bring them back to the homestead, walk to the water front (rapids for never freezing water), drop everything carried (which includes the carcass, of course), recover the breath if needed, skin, clean, butcher, tan.
When it comes to any predators, your dogs can get agitated and attack the animal even when it's trapped, so I typically tie my character's (single) dog to a tree before dealing with predators that aren't too dangerous (chasing a badger around the yard, for instance), but wolves and bear in the wild definitely should be taken seriously, unless they're breathless (at which time I can tie the leashed dog to a tree when dealing with the mostly neutralized creature). I'd also be wary about gluttons and wild boars.