Can i suggest you stop asking to babysit? They know you want to, they just don't need you to right now.
This is what i would do, find out from your son what his wife wants or needs, her favourite things, I'm not talking big things just brand of chocolate bars, sweets, crisps, bath soap, pamper stuff. Make up a bundle for mum but the key is to get things she definitely wants or needs. Things that hit the spot. Pop a few treats in for your son too, he is probably feeling exhausted and needs a little loving too. Mention to your son you'd like to drop it by on your next day off and ask him to be home. Say you don't mind if the baby is asleep, you'd just like to see them. (obviously you would prefer the baby to be awake but take the pressure off) Either leave a note with it when you go or say "just a little something because I know being a new parent is tough"
This should help in two ways; the first is showing you that you actually don't know your daughter in law that well - you have to ask your son what she likes. That might make you understand that she maybe doesn't know you that well either and that's why she may be less comfortable going to you to babysit. Disappointing, yes, but fixable with time. You just need to build up your relationship with her (not just your grandchild). Be around them as a family and your babysitting days will come.
The second thing it does is show the mum that you respect her and you are looking out for her, not just interested in the new baby. This should help put her guard down and by feeling warm and fuzzy towards you she may remember you more often. It may take time but take the focus off the baby and as a result you will likely find you see the baby more. The new mum is probably feeling a little vulnerable and just needs some kind words about her parenting, how well she is doing etc. Note that saying "he or she is thriving" is not the same as "you're doing a great job". The second is directed to the parent the first is directed to the baby.
I can understand that it's hard being a grandma and feeling a little left out but I bet it's unintentional and if it really isn't the above will likely help.
p s buy yourself a little treat too. Hugs xx