So the lack of plate issue is legally resolved with the document I posted a few days ago. Print that and save.
Now for the insurance issue. In NY a C class moped does not need insurance. If you bring a moped that is on the C list, then that becomes a moot point. Fortunately, the list is extensive, and completely wrong, allowing many mopeds that easily exceed 20.
I am going to pick from my fleet a moped that is on the list. This way not having insurance is also not an issue.
I have attached the list obtained directly from the NY DMV Technical Bureau in Albany. Print, pocket, and enjoy riding 100% legal with no plate or insurance in the Empire State!
As long as your home state has a reciprocity agreement with NY, these documents will work for you as well. Check with your local DMV.