Not sure on the hunter part but in my experience to butcher something killed by the military you need to manually set the butcher to butcher dead animals, for some reason military kills do not auto butcher.
In this case, even manual butcher jobs fail because it does not see any butcherable corpse.
In order to get hunting working at 100%, it appears to need 2 things:
A stockpile with both refuse and corpses enabled.
Surface refuse collection enabled.
If you don't have either of these things, the corpse will just sit on the surface. For some reason, once the corpse gets hauled to a refuse pile, an auto-butcher job fires off and it gets moved to a butcher shop.
Tried it several times by disabling surface collection or disabling the refuse piles. Unless both were active, nothing would cause the corpse to be butchered.
Without surface collecting and a generalized refuse pile, only about 1/4 of the hunted creatures would actually get hauled back by the hunter. Most times, the hunter would go eat, sleep, or on break right after a kill, leaving an inaccessible corpse (until I fulfilled both the above criteria). Once the hunter was done with that task, they would go back to hunting fresh game.
I'm going to start working on a bug report and get a save set up were the bug is active.
Also of note: If the corpse is on a stockpile, it will still be butchered even if surface refuse collection is disabled.
EDIT: OK, got a save where the bug is currently effecting a hunted corpse. Bug report is here
08181