If the child is seemingly capable of being potty trained but is not due to lack of motivation I would be tempted to motivate him by making him miss out on more playtime by being in a pull-up than using the potty. I'm not sure what this would look like for this child. Being part time doesn't help.

Would he be capable of changing his own pull up?

Maybe make him have a pull up change every 30minutes? Full out, lie down on the change table take it off, check if it's wet, put it on, redress. Anything to make it tedious. He will likely freak out through it but just keep saying, oh if you wear diapers at daycare this is what happens. It will be a HUGE pain for you but it might be enough to make him realize being in a pull up doesn't have any perks.

Oh, stop calling them pullups and call them diapers. They are a diaper in this sense as they are being used with the intent that he soils them...that is a diaper. Call them a diaper, as often as you can. Maybe consider switching him to diapers instead of pull ups at daycare.

Unfortunately for the 'average' child waiting until they are 4 means they missed the prime training period and they are not old enough and smart enough to just be lazy and to be difficult about it. It is hard to excite a 4 yr old with trying to sit on a potty. Now you just end up in a battle with a child old enough to refuse!

What province are you in? Will he be allowed in kindergarten? Here where I am they have to be full toilet trained to attend unless they have a noted medical condition/disability...that is usually enough reason to get these types of parents trying at home as they will end up paying an extra year of daycare if they don't train him.

Can you afford to lose this child? I would consider giving a probation period of a month or so and tell the family if he's not trained by then you will have to terminate. Just to get them working on it...but you risk losing them...if there is another provider willing to take on a part time almost 4yr old in diapers.

If he were full time it would be much easier for you as you'd have more time to do an intensive training.