I read this sentence as: Please exit from the toilet in the correct manner. Don't do any silly walks. Don't try to walk through the door before you've opened it. Don't scream, "She's gonna blow!" as you charge out of the stall.
It's a grammatical English sentence and even has a real meaning, but probably not the meaning the sign writers intended.
Presumably they meant something like please leave the toilet in the proper condition. You could arguably write this as "please leave the toilet proper," in the same way that you'd write "please leave the toilet clean." We do not say, "Please leave the toilet cleanly," because, as you say, the adverb cleanly modifies the verb leave, not the noun toilet.
Still, I don't know what the "proper condition" of a toilet is, so leave the toilet proper wouldn't make sense to me. Perhaps they mean leave the toilet clean and with the seat and lid down and make sure to flush. If so, that's not coming through with that terse message.
And even if the "proper condition" were unambiguous, and if they did mean leave the toilet proper, as has been pointed out in the comments, it would be more natural to use "leave the toilet proper1" to mean step away from the toilet (the apparatus) or else, exit the formal toilet area (and perhaps enter the makeshift toilet)
strictly limited to a specified thing, place, or idea the city proper
// the city proper