Your recourse is the document damage and withhold security deposit, as per your state's laws.
You can't withhold security deposit for a lease "rules violation" that causes no damage. If you have an issue with a tenant violating the terms of your lease, that's for you to take up with them during the lease period (e.g., grounds for them to change it or be evicted).
The more restrictive you make things, the more tenants will hide things from you. That's just human nature.
Your claim about the potential insurance claim is probably false and likely loses you credibility in the eyes of tenants.
A good reason to have pets disclosed is that it makes it easier for you to set reasonable policies and then see if your tenants are in compliance with your lease terms (e.g., # or type of pets, weight limits, etc.).
FWIW - I'm a frequent renter, past landlord, dog owner, and uber responsible dog trainer (so I see the gamut of responsibility/damage from dogs in many client homes). I've never minded giving a reasonable extra amount of security deposit when I'm renting, because I know I'm going to get it back with my style of pet management. I think "pet rent" as a per monthly charge is bullshit (pets are not people) & it's not like landlords can charge for young children, who often cause far more damage. The more landlords try to finesse the rules and allow "some" pets, the further you make rules that have nothing to do with damage potential, and the more you open the door for grey areas, tenant "lawyering" and people hiding shit.
My experience is that responsible pet owners can be great long-term tenants because they appreciate that not everyone rents to them & are thrilled to find someone reasonable who does. I understand landlords who simply don't want to rent to pet owners at all (esp if they have high turnover properties, allergies themselves, and so on).
Good luck with your next tenants!