You can say you decide to add rent was based on

their pets, plus you can ask for additional pet deposit. You can not discriminate people--whether they are married, big family, disabled...BUT you can definitely discriminate pets. You can refuse to rent to them because they have pets, not because they have too big of a family.

Keep record of all the communications, all responses should be in writing.

It's a learning process, sooner or later you'll bump into someone like that if you want to be a landlord. Keep learning, you'll be ok.

Good luck.