I feel for you, MaternityLeave. I understand that you must feel pressured, manipulated, and put in "second place" (my words) by DH to your ILs. Hugs!
You say you would like some of the MIL perspective, if possible, and as an MIL, myself, I think I can see it, especially in this case. Given her illness, I imagine that MIL is feeling a little desperate, afraid she won't get "enough time" w/ her GS before she dies or gets to ill to enjoy him. FIL and your SILs may be worried for her, as well. IMO, you are being very generous w/ your time, but no time may seem like "enough" to them right now and no location "close enough" (that would explain the house issue). Or they may be a possessive family, anyhow, IDK.
But, IMO, even if their behavior is driven by MIL's illness, that doesn't mean you and DH have to let them take over your decisions. In fact, I think it means you need to be even more vigilant about seeing that doesn't happen. You both need to make sure that choices about your life and your child are still made by the two of you and only the two of you.
What bothers me the most is that DH seems to have discussed the childcare issue AFTER he had already made a decision w/ you and even enrolled in a childcare voucher program at work. It sounds as if he mentioned it to his parents or sisters, and they objected. Instead of saying, "sorry, this is between ML and me and the decision has already been made," he made the mistake of listening and letting himself be influenced by their (so-called) concerns.
IMO, you need to have a talk w/ DH and let him know that, in the future, parenting decision need to be made by the two of you, only, w/ no one else getting a vote. If he wants to seek his family's advice, he should, at least do it BEFORE he makes a decision w/ you, not agree to a plan of action and then change his position b/c he spoke to them. But, given their ulterior motives, IMO, it's better if you discuss these issues w/ each other only. And surely, once you and he make your parenting choice, it should be final.
I get that DH's sisters would be doing more of the care than MIL, so her illness wouldn't interfere w/ that. Therefore, yes, a compromise of 1 afternoon a week might be a good idea.
However, I think you need to mull that over carefully, if you feel you can't trust MIL not to take him to see people who might endanger him. That's a common fear in today's world, and, clearly, you have a specific reason to be worried.
No doubt, you're worried that fighting DH on this would widen that "crack" in your marriage. But letting the ILs make all your decisions will hurt your marriage, too, and eventually, perhaps, destroy it. Please fight for what you feel is best for your marriage and your child.
In the end, resolving the issue by staying home may be the best option. There are many years ahead in which you can work, and you may truly enjoy being there for all baby's "firsts," etc. I know I did. Please let us know what you decide.