In one way, the problem is already solved; if MoG strops off, her place can be taken by the random she wants to invite, right?
But seriously, you have enough on your plate. This dispute is nothing you have to solve or get involved in at all. You've set the budget: the bride and groom control the guest list. End of. Get them to tell you when it's sorted and put it from your mind now.
This is for the groom to thrash out privately with his mother. YOUR DAD should not be drawn in. I'd be inclined, in his shoes, to tell her a simple no to the extra guest and if she insists, say she has to decide who gets the invite, her or the random, reminding her it's not her day!
Does she have form as a drama llama and wanting everything to be about her?
My own late FiL was an attention seeker who loved the drama of being coaxed and pleaded with. When our wedding was planned, I got loads of anxious traffic from DH's lovely mother that he was refusing to attend and might mean it. My family was so straightforward I hadn't a clue what he was about. I was a perfectly appropriate bride for his DS and there was no reasonable objection I could think of.
I just shrugged and said whatever he wanted was fine by me, he was invited, he was welcome and we'd get married whether he was there or not.
Turned out to be the smartest way to deal with his tendency to create needless drama.