Sounds like you have a good plan worked out. Sounds like a good compromise to keep things even.
I might disagree, (a very little), with the whole "You're a solo so you should just be happy we let you come with us" aspect of your argument.
I think it's different too that your odd wheel is a member of your family. Of course MIL doesn't mind sharing a room with the grandkid. My only thing is that in the OP's situation of course the two mothers with their six year old daughters love rooming together. I didn't take it that the third adult coming was a relative so I'm not sure staying in a room with 2 six year olds is quite the 'non-issue' that it would be for a relative of the kids.
LOL, what if the two women each had 3 kids and they were getting accomodations for 9, would the solo still pay 1/3rd? Same thing right?