California Wow
To answer my own question: the California Wow gyms have spin bikes and classes. The one on Sukumvit offers a few classes every day and has two different levels of classes.
I signed up for a one-month only membership for 1500 THB. The price could actually be less with some negotiation. As soon as I accepted the quoted price the salesperson smirked that little "gotcha" smile and I think I heard cash registers going off in his head.
Long-term memberships are said (on other IBBs) to cost between 850 and 1050 THB per month, so the price did not seem unreasonable.