I would start the new term as pay for the term upfront, no payment, cant start.
Dont worry about the sob stories, you will get lots.
People have to make a commitment, as you said, you still have to pay your costs and your fees will incorporate these costs.
Also specify that to run the class you need a minimum of xxx paid customers. If you dont get that number by a certain date, then the class cannot run.
People will always not pay, delay paying, give sob stories. It doesnt concern you, you are running a business/service for children.
Pay as you go is great for parents who dont want to commit their child to something, but you need to able run your club profitability and I would move away from this.
There is NOT ONE club we go to that we get a refund if we dont turn up. The only exception was swimming when DC broke her arm. Most clubs wont re-schedule either because again its a disruption for the organisers.
Dig your heels in, be firm, and it will work.
Good luck