How do I convince my patient that he's actually dead?

Unfortunately, this bug has been brought up to the developer, who acknowledged it was a bug and offered no help in actually fixing it. Other people have the same issue but nobody I've found has actually found a way around it. Hotfixes that have been released since the bug was reported don't explicitly report any fixes either.

I think your zombie patient is stuck there until the celestial powers-that-be come up with a workaround.

https://steamcommunity.com/app/535930/discussions/1/1737715419901973932/

While I haven't found a way for the patient to acknowledge their own mortality, you can resolve the services aspect by deleting the room they're holding up, and rebuilding it.

The zombie is still standing there, and I've found a few friends for him. But for the moment, none of them want treatment, and they just kinda stand there.