Success! I now have what may be the only legally registered post-2005 Morgan in the State of Massachusetts. The process is byzantine, but it is doable. It is the process to register "Replica" or "Kit" Cars (applicable for the small number of Morgan Plus 4s imported into the United States between 2005 and 2020). If there is ever another person trying to do this in Massachusetts, the steps are as follows:

1. Obtain a Massachusetts VIN (a "jurisdictional VIN) by having an inspection done by State troopers (it'll be a non-standard VIN with the prefix MA followed by five numbers).
2. Register and title car with the new MA VIN.
3. Purchase an old car (any pre-1974 off of Craigslist with engine at least as large as the Morgan you're trying to register) that has been operational and registered at some time in the last 5 years. Have it "junked" or destroyed at a junkyard (the junkyard will pay you for the car thereby offsetting some of the price of purchasing it). Have them sign the requisite state form that the vehicle has been destroyed. This serves as a sort of "carbon offset;" which is legally allowed to be used to get a lifetime emissions waiver for a "replica" car (the Morgan).
4. Get an inspection of the Morgan with the Massachusetts Motorist Assistance Center (MAC) bringing them the signed form that you have crushed an older car. They'll issue the lifetime waiver for the Morgan.
5. Get the annual inspection done at any MA vehicle inspection station: Now and for the lifetime of the vehicle, no emissions test will be done. They'll do the annual safety inspection and issue you the requisite sticker.

Crazy, but doable.