In general you only really need to return the perfect pelts, returning the carcass too just nets you additional money (and likely improves food in the camp, not 100% about that though). Occasionally there are some challenges where you're required to "bring 5 carcasses to camp" or something similar, in which case returning them is beneficial.
Pro tip: some of the smaller animals can actually be stored inside Arthur's satchel meaning you can potentially bring back multiple carcasses per trip to hit those goals.
In general I try to aim for a specific upgrade and focus on just the ingredients (pelts, feathers, horns etc) for that goal. Once camp is fully upgraded I sell everything to the trapper. He requires a tonne of perfect pelts for all the custom costumes and offers higher returns than the camp.
Hope that helps.