I work in a preschool/daycare and I have one boy in my pre-k class who is still potty training. So not all daycare centers are so tense about potty training.
But have you considered that while you are too busy to spearhead your child's potty training because of one infant, the daycare teacher has to try to focus on training your child as well as several others at the same time? And do you realize that it requires extra personnel to attend to children while one staff person helps change your child's pants? These are the things you would probably be paying for. And if you are looking for a center to train your child so you don't have to, then it will be a long, hard road because in my experience, if the parents aren't going to do their part, then it is like taking two steps forward and one step back. I wish I had a nickel for every time a child achieved a great trend in going on the toilet, only to have the parent put them in pull-ups as soon as they are home and all through the weekend. The parents simply cannot be bothered to do the dirty work. I am not saying you are doing this, I just want you to realize that the day care workers who potty train do not have a pleasant job. It takes alot of patience, extra clothes, time, and latex gloves.
Don't give up...keep trying. There are good centers out there that will work with you to help your child achieve according to his own good time.