Well - I think it is possible to do something with the best of motives only to find that the person on the receiving end has thoughts and concerns that you were not aware of.
It seems to me that this Mum has got her own concerns abut returning to work and that awful wrench of putting your child in the care of others - maybe she just needs to feel she has control of the organisation of the care in order that she can feel better about it all.
In fact, in retrospect, I guess you might conclude that some sort of family meeting to see who could do what, and what your DIL would feel happy with might have been a better way forward; but we all know how excellent the retrospectoscope is!
I know that on the surface this is about two kindly grandparents offering their services and that one could think DIL should just jolly well feel grateful - but, as parents, I think we all know that the care of our children is a sensitive issue.
I feel sure there will be a way of sorting this all out as it sounds as though you are all caring people who want the best for everyone.
I care for my GD one day a week while DD is at work (she does 3 days) and I offered to do it - I know that she is grateful for that and she respects our need to take holidays now and again (when she finds an alternative care option) - but for our part, we always respect her wishes, and follow the regimes that she feels are important (even if it might not always be the way we might do it!). Leaving your child is so hard, and knowing that the carers are respecting your views is a big bonus I think.
Good luck with all of this - and with your forthcoming surgery.
You are right that it can be a minefield, but it also carries undreamt of joys.