Please don't take offence at the washing of things. I do this, but only because my baby has very sensitive skin, and it seemed politer to just chuck everything through the machine with the levels and type of washing powder that I know is ok, than grill whoever gave it about what they washed it in, whether they used softener, how much powder they used.
I know it's something that even parent of children without sensitive skin can be careful about. So I don't think that's an offensive act at all.
You say that the other GM is pushier? Well I think that's your answer. Do you offer to have the baby? Do you want to commit to a regular date?
If I had two close relatives, and one offered to take the baby to give me a break, and the other sat back and waited for me to ask, it's a no brainer to see who would be the obvious choice. I wouldn't want to foist my children on someone who I wasn't positive wanted to babysit.
If you do ask, then perhaps try a more definite approach. Instead of the vague, "if you ever need a babysitter then let me know", try naming a time, "ddil, would you like a break while I take the baby to the park this week? Is Tuesday good for you? Wednesday then?" Or "you two should have some time together, would you like that? I could look after the baby on Friday evening if you'd like? They can either stay overnight at mine, or I'll babysit at yours and drive home later on. Whichever you'd prefer."
If the answers to those questions are "no" without a decent reason (and actually prior arrangements for that week is a decent reason, so suggest the week after if that's the case), on more than one occasion, then perhaps you do have a problem.
Is there anything that you do that they don't want around the baby? Specific things I wouldn't like are smoking and always bringing a dog with you.
I wouldn't want my baby to sleep in a smokers home as it hugely increases the risks of cot death. It also lingers on the person. If I had the choice between a smoker and non-smoker then again, that's a no-brainer. One increases the risk to my child.
I'm just trying to suggest possible obstructions that would affect the scenario. If there are things you can change to make this better, then in many ways that's ideal because it's in your control :)