J., I was starting to think, "Hmm, this lady should be more willing...." UNTIL I got to this vital statement: " I am on permanent disability for a reason due to a health issue."
Your son and his girlfriend are being selfish. Of course you'd like a relationship with these kids, but the adults -- not really acting or thinking like adults -- are basically blackmailing you: "If you don't take both, you can't see either, so there! If you don't take them you will never see them again!" Not mature. Not fair on you with the health issue. And as for your mom -- you and she must simply flat-out say that she never will babysit. She can visit while YOU babysit, sure, but age and health mean she should not be in full charge of kids this young. Heart surgery and they expect her to babysit?!
Be aware that son and GF may feel that your real, unspoken reason for not wanting to babysit the older boy is because he is not your son's biological child and is unrelated to you. There are grandparents out there who would, in your case, reject the older child because he is not "family." I'm assuming here that this is not your case, BUT you really need to see how your son and GF could assume the opposite and be thinking, "She just dislikes Boy because he's not Son's blood and the 'bad behavior' is just an excuse she uses...." Please consider ways to ensure that you are clearly making the boy as welcome as possible, and praising him when he's good, and spending some time with him. Not babysitting, just taking the initiative to see him, but not for hours. For a kid like you describe it sounds like a very, very busy, structured and occupied visit is best, where he is doing something every second and not able to get into his own trouble. Just be aware that though he does have these issues of behavior and others see the same problems you see, it will come off to son and GF as if you are just rejecting the boy because he's not Son's child -- especially if you say you are OK with taking the baby who IS son's biological child. I'm not saying here that YOU feel that way, but it is very, very likely that is how the parents will see it, no matter what you tell them.
If you want real backing for not sitting as a regular "job," go to your doctor and ask if your disability precludes it. Your doctor may confirm that your health would be affected. (If son doesn't believe you, you can then legitimately tell him what your doctor said.) And if you are drawing disability pay, could that actually be jeopardized if you are seen to be looking after a rambunctious toddler and a needy infant two to three full days a week? That's something to consider as well. It might be that even an unpaid babysitting arrangement would be viewed as meaning you are more able than your disability arrangements allow.
I would tell them you love both kids, but cannot babysit them because of the disability, period. Then I would invite just the older child on an occasional special (and short) outing, or find a six- or eight-week session of a once-weekly class you and he do together for an hour.
When you say your formal no to sitting either kid, you could at the same time offer the parents a list of lower-cost care options -- do some research to present to them. In other words, keep the doors open and show that you do care about the kids and want to help them lower their child care costs, but you are not an option as a sitter.
I would not take the baby and say no to taking the older boy, though. This is what your son means when he tells you they're brothers and you have to take both or neither -- your son is already feeling you are rejecting the older kid and not because of behavior, either. But I think your health has to come first and sitting just the baby won't help your health. Say no but stay involved on your own terms.