They want to invent a safety reason (both airways open), but then contradict themselves when saying a noseclip is easy to take off. This tells me that the first safety reason given is a fallacy (lie). I think it's an excuse to ban diving. This is most likely to ban perverts from enjoying the underwater view of women.
Mask or Noseclip do not make much of a difference in regard to drowning or the rescue.
I'm afraid that the owners of the pool are irrational, and negotiation with management only will have them alarmed. With the pool staff, lifeguards on the other hand you might have a chance on creating a good rapport/relation with the them and the fellow pool users, so they will allow you (look the other way) to enjoy your dives with a mask.
Using a fully transparent Sphera (looks innocent) I think works best. Your excuse can be that chlorine in the nose is irritating and was not allowed by your doctor. A noseclip is not sufficient as it does not allow your mucus to be drained, creating infections. Both are real and valid reasons.
I hope this helps and you can enjoy the local pool water.