If I was in your situation I would lie and say I didn't have a pet, sign the lease and smuggle them in. Put their crates inside cardboard boxes to carry them in. Then after a few weeks approach management and try to get permission to bring your emotional support cats in.

Worse case scenario you've bought yourself some time to save up to find a better place. Just make sure that if they find out to start looking for a place as soon as they give you notice to move because it's super hard to find an apartment with an open eviction case. Look into your state's laws, most states they require longer notice periods when the eviction isn't for non-payment of rent.

Normally I'd say be honest from the start so you can find a landlord that will accept them, but it seems like you're in a tough situation right now.