Surely if your new rental will accept cats, why can't you take all your cats with you ?

If you want to take all your cats to your new home.....

Try offering an additional cat bond in case of damage to the property directly to the landlord.

Get references from past landlords and / or vets saying how well behaved your cat(s) is.

Offer to steam clean the carpets when you leave the property.

Properly trained cats with careful owners do not ruin rental property.