I would stop. It's so ridiculous how people get over potty training! (I haven't read any of the comments, so this is not meant towards anyone who has left a comment). I just read today that the average age for boys to potty train is 37 months old.
In my opinion, it's a WASTE OF TIME and ENERGY to force a child to potty train. When you wait until they are ready, it's a piece of cake! It's so NOT true that the longer you wait, the harder it'll be to potty train them.
It's important to not make it a big deal...to NOT stress him out over it. It's just potty training. I think your gut is telling you what to do about it. Don't do it right now.
I potty trained my oldest when she was 3.5 years old. She was the size of a 2 yr old (literally) and so I never felt pressure to potty train her. It was quick and easy when the time came.
Then when I had my son, I felt pressure to potty train him around age 2. So, I did. He was potty trained for four months before I decided to stop. It was just way too demanding on me. I had a newborn, and my son didn't know how to pull his pants up or down, and he needed regular reminders, etc. He'd have to go pee at times where I was taking care of baby and couldn't stop and he'd have an accident because I couldn't get to him in time. So, I stopped and tried again a little after he turned 3 years old. It was a breeze. He didn't seem to get it until I had him go bare bummed. Once he was bare bummed, he potty trained in just a couple days, only having a couple accidents. That has been the easiest way to potty train in my experience. They pay more attention when they have no pants on...hehe.
Then with child #3, I didn't even try to potty train her before age three. She had times where she wanted to go, so I'd take her. But usually she lost interest fairly quickly, so I stopped. She just barely turned three, and a few days before I said "You are almost 3!! That means you get to go potty on the toilet and wear shorties like your brother and sister! That means you are going to be big enough to do that just like them!" I made it super exciting, and she felt super excited about it. I've been potty training her for maybe 6-7 days, and she's had 2-3 accidents and that's it. She's starting to feel quite reliable about it all and if she keeps it up, in the next couple days i'll feel content she's fully potty trained. Most days she doesn't have an accident.
When they are ready, it's SO MUCH EASIER.
I would feel stressed thinking of daycare pushing it before he's ready. I know they probably don't like having to change his diapers, but you are PAYING them to take care of your child. I don't feel like it's their place to tell you what you need to be doing with your son. I understand it affects them, but something like potty training can be too stressful to kids, and it's something you should have a say in.
Anyway, I'd give it a break if it's getting so upset about it. That shouldn't be part of potty training.
Good luck!