I have 2 rabbits that do a sort of cough. One does it often after drinking so I think it’s just due to the speed he drinks. The other has had some health challenges though. He does what I describe as a gag/cough. Ive mentioned it many times to the vet but we only end up speculating on the cause. He does have affected facial muscles so one side of his face is pulled up and he gets a bit wet under the chin. He manages to eat well though so his chewing and swallowing does not seem to be affected by the spasm. I do wonder of he produces more saliva then the norm and part of it is his continually swallowing that down? Or if he has something like post-nasal drip if that’s possible in rabbits.
He does have sneezing episodes but no discharge.
If your vet does find there is more saliva in your rabbit, that might be a clue to some dental issues. Or over-active saliva gland perhaps. Another thing that could happen is if there are some molar spurs that are hooking food at the back of the mouth and triggering a gag or cough.