We put our kids in Sunday School and had a similar situation when my daughter first started going a bit after she turned 1.
When it came time to leave her in the classroom, I did not go in. I didn't try to get her accustomed to the place with me in it. We DID ask a nursery volunteer to come visit with us outside of the class for a while. Then we went to the class, signed in, and the volunteer took Becca into the room. Becca, of course, started crying and getting a little panicked-looking. The nursery had a 10-minute cry rule (after 10 minutes of crying they would page a parent) but I told them to call me after 5. I have never been called; she always stopped crying just minutes after I left.
Looking at her when I first left I wouldn't have thought she'd quiet down so quick, but she did. If you've never tried it before, I suggest you see if your boy would calm down after a couple of minutes or if he would continue to panic.
I do work in the 2 year old classroom at my church every couple of weeks. Some of the parents there (who didn't have their kids in the Sunday School until they reached this age) employ the approach someone else mentioned; they will play in the room with their child, and extend the amount of time they are gone from the room a little each week. It works for some toddlers and not for others.
With our second we started him in the nursery pretty young, and he has never had problems going in there. Hey, it's a familiar place with lots of toys and cool friends, what's not to like ;-).
Um, to make a long story short; I think you should see if you can find a way to adjust your boy into the daycare setting so you can get your exercise time. It is probably worth a little fuss and hassle to see if you can get things transitioned.
Mama, homeschooler, midwife. DD (13yo), DS (11yo), DD (8yo), DD (3yo), somebody new coming in November 2013.