Wow. Poor little guy. He's probably very confused now but I can understand your situation and need to coordinate the naps.
Are you working and that is the need for the sitter? Does your son normally nap at home twice a day at the same times of day?
Considering the sitter has three children of her own and a few others as well as your son under her care I think she may have too many children to care for at once. That's another issue.
Anyone have any advice to share?
I disagree that she has too many kids. Most people can easily handle six to eight kids without any issues. In fact under six kids is harder than over six.
But, I think they can work this out if the daycare provider is willing to try a few of your suggestions. I have a daycare, and ALL of my infants, even the co sleeping infants sleep just fin in a pack n play alone. I haven't ever had a problem with it. However, I don't let them cry for 20 minutes either. I think that's what is wrong. He's old enough to be on one nap a day, and lay down with the other kids when they lay down. It might take some transition time, but if she sets things up so they are all sleeping together at the same time, and the non sleepers are in another part of the house, it should work in a few days to a week.
It CAN work, but they way they are doing it now isn't working. She just needs a new plan. Maybe all of you can work on the same thing at the same time.
It's harder because he's older. He's only known one way, and he thinks it's the only way. It would have been easier if a daycare provider had started it when he was an infant. But, he can do it now, he just needs a new approach.