Just bought M&M 22 Compact with the threaded barrel. It came
with the adapter and a thread protector for the ADAPTER. It did
not come with a thread protector for the BARREL.

The book says (and you can plainly see) not to fire the pistol
with the adapter removed. I want to put a BARREL thread protector on
it and use it without the adapter.

EWK ARMS makes the barrel protector and a "Recoil Guide Spring
Centering Bushing". It looks to me that the barrel thread protector would probably be OK, but for under $10 the centering
bushing would make sense. Has anyone done this or have any
thoughts about it? Thank You.

Thats because its just as easy to run it with the adapter and the thread protector on the adapter. Ive personally never ran mine without the adapter. I've just never seen a point in doing so.

If you are looking for the "original" thread protector that should have come with the pistol, just call S&W customer service or send them an email through their website. They have always been VERY helpful to me and will often send replacement parts to you for free.

Several years ago, I had a 22A that needed a new recoil buffer (little nylon disk) and they sent me a package of 3 at no charge.

Just a few days ago, after reading that some people who've bought the M&P 22 Compact didn't get a wrench for the thread protector/barrel nut, I decided to get out my box and check. Sure enough, I didn't get one either so I sent them an email requesting one.
I got a response yesterday, saying they were sorry for the inconvenience and that they are sending me one right away.