I have always asked new acquaintances whether they are allergic or object to cats when I invite them for the first time.
If they are allergic, I suggest we meet outside our home, as however carefully I vacuum there are bound to be cat hair around.
If people aren't allergic, they are welcome to come, and we put the cats out if the weather is good, or into another room for the duration of the visit. Obviously, a cat-hater can't very well stay the night here, but otherwise within reasonable limits they are welcome.
Pet owners with a very sentimental attachment to their pets simply do not understand that others do not want to be visited by pets, but by their owners.
It's your house OP, so your rules, but it may well cost you the friendship.
Personally, I find it very inconsiderate of people to inflict their pets on others, but that is just me.
A dog that has not been trained to be alone at home for a few hours, probably cannot be left without whining, soiling floors or destroying furniture, but that is no reason I can see for OP or others being forced to admit the animal to their homes.