V.W.
No, you are not being ridiculous! Not at all.
She is just not using sound judgment, and from what you describe, sound judgment is not her forte.
You have every right (and responsibility) to protect your young children from strangers. And just because your mom has dated the guy for 2 weeks, or a month, or whatever, does not make him any less a stranger when it comes to your child. Your children don't need to have close relationships with these men... nor ANY relationship with them, imho. If you are hosting a get-together at your house, I don't see why your mom couldn't bring him with her, to be introduced to your children only as "grandma's friend ___", but the children should not treat him as a member of the family... and that sounds like what your mom is really after. If she introduces him and behaves the same with him in front of your children as she would say... a female friend of hers that you don't know well... then I see no problem with that. But if she wants the kids to sit in his lap and hug and act like he is their grandfather... then no.
I don't know how to keep your relationship from suffering. She sounds like she is not willing to be reasonable, so unless you give in then you will suffer the consequences of her not getting her way. Very sorry you are dealing with this.