Tell her no.
In fact, tell her nothing, get your DP to do it.
It's his mother.
He can tell her that you can't have two badly behaved dogs in your house, as it will be dangerous for your cat.
He should also tell her that in order to ask a favour from someone, it is usually necessary to be civil to that person, and not treat them in a rude, hurtful and ignorant manner.
She sounds like a nob.
And a spoilt brat.
She is a spoilt brob.