Ok, first some background.
My first suggestion is have DH deal with this. Next have him ask to meet the new GF before the party. I agree its not far to have Liam b-day party get turned into meet the GF party. But just ask to meet her before hand. As for MIL feelings, I'm guessing they have been divorced for a while now, and its a fact of life, that he has GF's.
It stinks that it will be the first time you meet her, but I personally wouldn't tell my FIL he couldn't bring his guest.
Thanks Vanessa for responding so quickly.
Actually, we sent FIL a email asking not to bring the gf out of respect for us and MIL.