I suspect though the issue here wasnt just the non holding of the baby but the general feeling that she wasn't wanted. This is an inevitable problem when international travel is involved as very few women want their MILS actually staying for a whole month, especially when they have a young baby. Would the OP have welcomed her MIL doing this when she had her first baby?
OF course its really difficult when the cost is high, and she has the added fear of flying. Having got there she wants her 'moneys worth' as it were, but I still suggest she restricts herself to two weeks on the next visits. Two quality weeks where welcome is better than four weeks outstaying your welcome.
Another point might be to try to make at least one of the two weeks when son is off work, and when at home alone with DIL to try to give her some space, take GC out certainly if she agrees, but otherwise go out for a couple of hours to give her a little space, a walk, a visit to the shops, any local atttraction, or at the very least to sit and read a book in the garden, not to be constantly hovering around her and GC.
Bought the wrong Hot Cross Buns 😩
In light of our current weather.......
How much do you spend on yourself?