I don't see the OP compromising in anyway. In fact going out of her way to make things difficult.
Would it REALLY be so difficult to call GM a couple of weeks before the birthday and say 'Hi MIL, it's DGS birthday in a couple of weeks, I am taking him out with some mum friends and kids during the day - but we will be back by 5 if you want to pop over for a birthday tea.. ? Is it so unreasonable for a GM to want to visit on the day. Especially if she is prepared to do an 120 mile round trip ?
There is no way I would NOT expect GM to visit on a second birthday.
When they get older and have their own mind to do things then that's when things change. But he is TWO fgs and doesn't know what day it is !
Yes of course YOU want her to come on a different day. Because YOU don't want his grandmother to have the pleasure of seeing him on his birthday.
This has EVERYTHING to do with you and MIL awful relationship and an inability to see that your partners mother wants a part in his life.
Yes I bet she does turn down your 'alternative offers' because I bet they are NEVER made for a significant day. .A few days before or after a birthday... Never on her birthday, but 'sometime later' . I think it is YOU that is determined she will never be able to have 'significant' time with him.
Does your own mother never see you on your birthday. Does she get to spend Christmas , his birthday or her own birthday with her Grandson or is she also relegated to an fee days later' ?
Continue in this vein and I won't be surprised if your son treats you in the same way when he's older.. firmly at arms length. I think it is you that is trying to 'control' things OP. Not your MIL.