I think I would go with something like this:
"Gosh, MIL, we certainly don't mean to make you feel excluded from Son's life and are surprised you felt that. Not our intention at all. If somehow it seemed that you weren't welcome to cook or host family dinner with/for him, that's not what we meant - of course that would be more than welcome in the future, if we just coordinate to organise a time that suits both us and you.
Of course you understand that we, as the parents, will make the plans for the day of or 'official' celebration of our son/children's birthday(s). As Son's parents, we have our own ideas for the day of his birthday this year, which happen to be just with us. We don't think it's excluding any particular side of the family with our plans as no side is involved over the other. Actually, I think if we were to have the family meal you suggested at your house on his official birthday, it would exclude my side of the family from official celebrations, which would favour one side and we don't want to do that.
By not having anyone else there on the day, we are only thinking of having a special day with our son ourselves, and certainly not trying to exclude either side of the family from his life. You could still celebrate his birthday by having a cake and candles another day for him to mark it with your side of the family too, if you choose! My side of the family might do the same."
Long and certainly not as sharp and pointed as is deserved, but I think gets the point across without compromising on it and (hopefully) while acting as a reminder that you're being really reasonable and she isn't. In the interests of maintaining harmony while not being walked all over and all that.
Also long because I am procrastinating again.