It doesn't matter. Your response should be the same. 'Silly grandma - little girls live with their mummies and daddies!' And laugh.
And maybe try a few more boundaries - 'oh, no, sorry mil, we're having a family weekend so it's not convenient for you to stay'. If she lives 120 miles away, I'm slightly baffled by exactly how much she CAN be interfering though, tbh. It's not like she is there every night when you get home. More like it's supremely irritating when they come over on high days and holidays?
In which case, sure, it's annoying, but don't over-react.
By taking her ridiculous suggestion seriously, with po faced letter writing and all the paraphernalia discussed above, is giving it far more weight than it deserves.
Laugh at her ridiculousness, and live your own lives.
Any suspicion that you are taking this seriously runs the risk of, you know, you taking it seriously. Which is bonkers. So don't. There is no chance DS will live with mil. So don't waste your time by investing any more of your life fussing about it. Laugh at her, to her face if necessary, and move on.
Your dh is in some way right - he isn't taking his mother seriously - therefore there is no reason to fuss, bother, and cause angst. Dd isn't going to live with grandma. Ha ha ha. Silly grandma.
And sort your own birthday parties out. She can only plan it if you let her. It's easy enough to say 'oh, we've already organised it!'