My dentist isn't registered with Invisalign. What should I do?

Sorry to hear that you've gone through this, but I would recommend contacting your dentist to inquire about this apparent misunderstanding.  I know that you must feel disappointed, but without more details from both sides, I can't make a definitive recommendation other than, your dentist will probably be willing to talk with you and work things out with regards to this issue.

It’s time to ask for
a refund, and contact your dental regulating agency. What does your treatment
plan say that was signed before treatment was started? Ask Invisalign for recommendations
of providers in your area.

I feel badly for you.  You need to look at what your treatment contract said.  If you were paying for invisalign and did not get it, then you can ask for a refund.  There are other clear aligner "invisalign-like" systems.  Ultimately if there has been any misrepresentation and you can't resolve it you can contact Board of Dentistry in your state.