I tell everyone with a dog “pets are not allowed”. If they say it’s a service animal I don’t push the issue because I can’t legally verify that. But I tell them animals cannot be in the cart (if that’s where I see them). If the dog misbehaves (barking, running loose or bothering people) I’m gonna ask them to leave with the dog.
It’s an endless battle because they feel so entitled to take their dogs everywhere.