I hear ya !
I am also transitioning a 13 month old and he is so clingy that I feel suffocated. As soon as he is not touching me or being held by me - total meltdown as though it is the end of his world.

It is wearing me down and I am in week 3.

Yours sounds much harder considering it is part time and the mom is on a totally different page.

I would probably cut my losses in your case because him transitioning successfully and sanely when mom has him half the time and is giving in to his every want and need makes it hard for him to get used to being 1 of many kids. It is tough when kids are the centre of the universe at home and then come into a group care setting where they have to share the adult and go with the flow.

I am considering giving notice throughout the day -- well ...I fantasize about it but I think I will probably be okay in a few more weeks I hope yours will too but I would at least talk to the mom and tell her how hard the adjustment is for her son.

It may be the mom's guilt causing her to do all this attachment parenting and she doesn't realize how much of a shock it is to alternate between one extreme to the other for this guy.

Good luck I sympathize