NRaas Register's Animal Control settings governs strays. It's not perfect, even when set to zero for a particular species some strays will still appear from time to time, but they won't hang around for very long.
NRaas StoryProgression can be used to set a very low cap on the resident cat population. You can't set it for zero because to the mod, a zero setting there means no cap. The mere existence of SP will shut down the Neighborhood Pet Adoption scenarios before they happen and Overwatch has a similar setting. SP can also be used to disallow adoption outside of the Neighborhood Centers, but unfortunately that covers human baby adoptions as well in the same step.
I don't think there really is a solution to forcing the game be unable to produce or simulate a single cat anywhere at any time if the existence of just one or two here and there is that distressing to the player but they do want dogs in their game. Other than, sorry, playing without the Pets EP.