The relevant part of the "Possession" quote needed to answer this question seems to be:
The possession lasts until the body drops to 0 hit points
Again, given that we are assuming that the answer to your other question is that the mouse is still possessed by the ghost, then the question becomes: does "the body" in the quote refer to the currently possessed body (i.e. the mouse) or the originally possessed body (i.e. the barbarian)?does "the body" in the quote refer to the currently possessed body (i.e. the mouse) or the originally possessed body (i.e. the barbarian)?
Obviously, if the former, then killing the mouse will end the possession and Conan, now returned to his original form, can give the ghost a good whack for possessing him. If the latter, then since the originally possessed body is still not at 0 HP after turning from a mouse back into a barbarian (i.e. Conan is still conscious and able to take Actions and such), then the possession won't end.
It depends on the interpretation. I'd say that, since no such distinction is made, and no mention of conditions like Unconscious or Incapacitated are present, it makes sense to reinterpret "the body" as "the current body" as the strictest reading, in which case killing the mouse would end possession, since the mouse's HP was dropped to 0killing the mouse would end possession, since the mouse's HP was dropped to 0.
Saying that, I'd argue that this might go against the spirit (no pun intended) of the possession effect, so I'd say that the barbarian would beremain possessed regardless of what form it takesforms he switches between, and only the barbarian reaching 0 HP in his original body would count with regards to ending possession, but that'd be my ruling, not necessarily what the wording of Possession says.