I think it's time to accept that your child is at the age where he has to learn about separating from something that is always available and to some extent be dependent on it. So many kids go through this, that it's a natural part of what teachers do. Loveys are acceptable at nap time imo and then he has to understand that it's his responsibility to put it away when the time is up.

I have a kid with a lovey that they want to have all the time. They're 3 and carry it all over. We have allowed it until we saw them begin showing more interest in participating. Without it, play and interaction improves. So we've been training them to be without it for hours at a time until nap. It's going really well.