Well, thanks exasperatedjoe, but I feel compelled to point out I’m actually in the camp of “one more isn’t going to overwhelm the bowling alley.”
To me, the woman has said, “If that’s a problem we’ll just come and pay for ourselves.” Perfect! They pay their own way, and if there’s not enough of anything, then you just say repeatedly to the mother, “I’m sorry there’s not a goodie bag for little Portia, but we didn’t know she was coming”; “I’m sorry there’s not enough bowling shoes and little Portia will have to rent her own, but we didn’t know she was coming;” “I’m sorry little Portia doesn’t have a name tag but WE DIDN’T FUCKING KNOW SHE WAS COMING BECAUSE YOU NEVER RESPONDED TO THE FUCKING INVITATION THAT SAID ‘RSVP’ IN BIG FUCKING LETTERS! Um, I mean, ‘we didn’t know she was coming’.”
IOW, you can subtlely reinforce to the clueless mother that to the extent her daughter is missing out on the specifics of the party, that is entirely her fault. Other than that, cake and pizza can be cut a little smaller to accommodate one more little stomach, and both your daughter and the other little girl will be glad if she can come. So, yeah, relax just a skootch. That doesn’t require you to believe the mother’s handling of the situation was okay; it wasn’t.