I don’t know. Muslims invite people hoping to convert (’revert’) them. I have actually never been to a mosque.
Although Muslims do not have dogs as pets, they can have dogs for hunting (i.e. functional). My guess is that a service dog would be regarded as functional but maybe not in a mosque(?).
I recall there was ain issue with service dogs in the taxis of Muslim drivers (maybe in Detroit?) a couple of years ago. Just made me wonder. I don’t suppose a church or mosque or temple is a public place in that they probably can make rules about who and what can come in there. I was just curious.