I doubt they only have pedals accommodating cycling shoes because not a lot of people are going to try out a class with an $80-$200 price tag (for the shoes) only to find they don't like it.
Shoe fit is really important---and from my experience more important in a spin class because (at least for me) I was riding high to higher intensity the whole time, so hot spots under and on top of the foot which wouldn't develop in road cycling showed up in the spin class. As twice, or go look at the bikes.
Alternately, you could bring your own pedals. Pedals swap out as a rule with a 9/16 or 15 -mm flat wrench or with (I think) an 8-mm allen key from the back.
I really enjoyed spin class---would still do it if my schedule allowed.