Firstly my sympathies with the menopause, do insist on a referral to the local menopause clinic so you can get the best help, and realise that you need to care for you too.
Secondly your GS's behaviour is entirely normal for his age, but if you can you need to manage this between you, with DiL being firm, prising him off her leg, giving him a cuddle with reassuring words that she will be back soon, and then handing him over to you and leaving straight away. If she makes a meal of goodbyes, then your GS will really think there is something to worry about and the fuss will continue. He will grow out of it, and I'll bet that once she has gone he settles down to play. It's all good practice for when he starts nursery or school, when he will feel the same anxieties. Please don't worry that it is anything to do with you, it isn't.
It might help at a calmer moment when she isn't stressed, to ask how she is coping, tell her briefly about your struggles with the menopause, how it makes you super sensitive, and that you need to feel you are both on the same side, working together to give her son a happy childhood. If you can, try to find a reason for asking her advice and following it, she may need reassurance too
Good luck, it isn't easy, but in no time at all he will be at school, and you will be wondering how to fill your time!
I have learnt that you do have to tread on eggshells with DiLs, and if you are going to buy baby equipment for your own childcare times, then either buy exactly the same as she has, or ask her to suggest a make or model, or come with you to choose.