For what it's worth, my MIL used to do call my DD "her baby", take her to play in a separate room from me, etc. and I just gritted my teeth - I'd always had a good relationship with MIL up to this point.
However things deteriorated, and the final straw was my DDs 2nd birthday. For her 1st birthday, MIL made DD an exact replica of my DHs 1st birthday cake. Fine, very sweet. She can have two birthday cakes. For DDs 2nd birthday, yep! MIL made an exact replica of DHs 2nd birthday cake - complete with the original 35-year-old plastic decorations.
At which point I announced to my DH that DD is our daughter, not a reincarnation of DH. And however much MIL loves her, she needs to find some other interests as well. And this year I am buying a "Hello Kitty" birthday cake for my DD as that is what she has asked for. And if MIL brings an unasked for cake again, it is going back in her car.
Sorry, none of this is strictly relevant but it made me feel better!
Anyway, I'd say something now before things get worse. They are 300 miles away, hopefully it will then all calm down before you see them again.