The easiest way to do it is probably not what you have in mind, and also probably the most expensive.
I seem to recall some of the manufacturers of AR15 lower receivers offered custom serial numbering. If you don't want custom, just sequential, that is probably easier. You would have to order them, have them shipped to a local FFL.
You could then assemble it with a dedicate .22LR upper.
If it was me, I would go with Big Shrek's name plate idea.