Sprinkle in some 4th in your 5th - available for all types of flair.

I didn’t find anything that would prevent me using my bite vs a freshly killed creature to gain some health back?

Classification of corpse != creature.

The only point I did find it that it said creature, but a dead body is considered an object?

Because 5e doesn’t want to bother tracking an extended time after which a dead body changes from creature to object while perhaps counting as both, so ANY point in time is as good as ANY other - but placing it at the same time as their death doesn’t need additional accounting.

to gain some health back

Carry a bag of live rats, chomp on them. Same effects, slight investment to make mechanics work.