Some of my students cheat, which can get their accounts flagged or even their IP addresses blocked.

Then they can’t participate in our class league, and their parents may end up leaving the chess school. Recruiting new students is already very difficult these days. What should I do?

tell them not to cheat

Tarring and feathering.

There are many threads about cheating. Now we can get some unbiased information.
Can you give us an estimate of what percentage of your students cheat? It will give us an idea of how common it is.

@deepbluecanada

I would never reward them for cheating. Having said that, I don't know if it's time to report them. That's an ethical question only you can answer. I'm not a teacher, but I'd like to think I would act responsibly if my students cheated. Some people might argue for you to show some integrity (in this case, parting ways with the students and reporting them, perhaps). I'm not trying to be cruel. If your financial well-being rests on keeping the students, maybe you can give the cheaters one last warning (but don't bend on this--if they cheat again, they're out). I hope I'm not coming across as "holier than thou." The other thing you have to worry about is your reputation. Think carefully before allowing them to continue as your students.

@deepbluecanada said ^

Then they can’t participate in our class league, and their parents may end up leaving the chess school. Recruiting new students is already very difficult these days. What should I do?

Well you can't blame the platform, cheating is not acceptable by any means

@deepbluecanada said ^

Then they can’t participate in our class league, and their parents may end up leaving the chess school. Recruiting new students is already very difficult these days. What should I do?

If they cheat they don't want to improve. Let them go...

For a fair play boost: Making the students thoroughly analyze their games might make them feel that they would never have had to cheat; having to write appeals is also a tool to make people think of their actions (even tough I share the believe of some European jurisdictions that incriminating oneself is strictly against human nature).

To unblock flagged accounts: Do not write a single word on the forums because it will not help to lift bans anyway, just send the students in question to lichess.org/appeal while they are logged in to their marked accounts.

To make your cheating students play on so you can keep taking money from their parents or guardians: Set up your own private chess server on a freely hosted website and additionally run a chess bot on it so that it can run permanently.

Cheating is of course a problem of character. You have to make this clear to them and maybe it is a good idea to tell them that their parents are paying you for teaching them and for showing good results. Their parents do not pay you for results that are achieved by cheating. Essentially your students are cheating their parents and of course themselves. If they do not stop you will report this to their parents and you will stop to accept them as a student. If this does not help, then the students are scumbags and you should not bother.
Another way to handle students that do not learn is to explain to them that you do not care, you are doing your job, they do not learn anything by cheating and if they are on their own in OTB play, they have a problem at hand. But this is not your problem, but theirs. This should make them think.
You are in a tough situation and you have my sympathy.