First, I get where you are coming from when you say you like that he doesn't want to be with other people. However, you are on the verge of creating a "monster" for lack of a better word. If he feels that you want him to be clingy, and he likes being clingy, it will just get worse with age. Do you and your husband have date nights? Do you leave him with a sitter? I am a preschool teacher and I see this sometimes with kids that are never, not with mom/dad. They have meltdowns, shake, & can't play with other kids because they are so worried about mom not coming back. Trust me when I tell you that you are not doing him any favors by allowing him to be with only you. A class where you can wait outside without distracting the teacher is what he needs.
When my kids were babies, I did in-home daycare. I was with them all of the time. However, my husband and I had a date night almost every single week. I needed it, and the girls needed it. Sometimes there was crying when we were going, but they got over it. Since we did it pretty much every week, they knew we were coming back and looked at the sitter as a new play friend.
You are doing the right thing by signing him up for a class. He needs to slowly get over being with mom and dad all of the time. This is not healthy for anyone. Most kids do fine with a little class here or there. Those that don't, truly benefit from being exposed to small classes here and there. It builds their self esteem and a little independence, and that is a good thing. Most kids go to preschool now, and those that are confident that mom/dad are coming back to get them, do really well and can consentrate on learning. The kids that are insecure have such anxiety that is tough to watch and sometimes tough to deal with. They can be very distracting to the entire class. Then their are other kids that meltdown and mom caves in and takes them home....those kids are controlling. These kids will continue to control until Kindergarten, then it becomes "sorry you have to go to school and all of the crying in the world will not stop me from leaving".
My daughter had a little boy in her Kindergarten class, (never went to preschool & was never without mom or dad) he cried every single morning. When I tell you every single morning, I am not kidding. This boy screamed for his mom not to leave him, and would throw himself on the ground. At first, we all felt bad for him. By the end of September, the rest of us dropping off our kids, we were done listening to him. By October, his mom was so done and at the point of anger with him. He started getting better right before Thanksgiving, still crying but not screaming. Then we were off for break and he was back screaming. Then after Christmas break he was back to the way he was at the beginning of school. The only person I felt sorry for, was his mom. This poor woman looked like she was at the end of her rope.
Did he immediately meltdown, or was the class too long for him? Maybe you could look for a class that is not an hour long. Most park districts offer "all by myself" type classes for the young ones, that are less than an hour. Just the perfect amount of time (30-45min) of time without mom.
If you really don't want to send him to another class, start with a sitter that comes to your house and watches him a couple hours a week. Ask friends and neigbors to recommend a sitter and try to use her at least once a week. If he is at home, it might make the separation easier and then once he knows that you are coming back and not going to leave him, he will be ready for a class. Kids are smarter than we give them credit for. If he is controlling you and getting away with it, he already knows exactly what to do to get what he wants.... you. :) Chin up he's young, just a little encouragement from you will go along way. You need to be okay with leaving him first, then he will be okay with you leaving. :)