It is not difficult to incorporate. If you search online, you will find a list of disposable email services, for example on GitHub. As disposable email services advertise themselves as such, it is not difficult to build and maintain list.

Our service included feature to disallow disposable email addresses from registering. We typically sent an email validation, if emails were returned or not responded, we will flag those services to be manually checked and added to do not register list. We eliminated over 95% of registration with disposable email.

Solution doesn't need to be perfect in elimination, just reduce the problem enough so that it can be handled manually.

That's pretty much an impossible to win game of cat and mouse

Impossible for perfection, but a solution that is 99.9% effective would still be a viable solution. I do it manually with people who sign up for our test system.