It it a reasonable thing to do or is there something illogical about it beside money?
An 8-round full-auto seems to be a bit illogical, but that's up to the user, I guess.
Not counting the gun, how expensive is the licensing and gunsmithing?
It is perfectly legal to build a full-auto firearm or to convert one to full-auto. However, that work must be done by a Special Occupational Tax Class 02 manufacturer (who must first hold a Type 007 or 010 Federal Firearms License as a manufacturer).
Also, and more importantly, that gun can never be owned or possessed by anyone who is not a Class 02 manufacturer, Class 03 dealer, or a government agency.
In other words, as a "private citizen" you CANNOT do this. Since
1986 the registry of full-auto guns in private hands has been closed, meaning that you cannot register a new "transferrable" machine gun. You can buy, sell, trade, etc. those in the registry before that date, but cannot add one more to that list. That's why the prices are so astronomically high. Lots of demand, very limited supply.
The costs to set yourself up as an FFL manufacturer and SOT 02, including paying the license fees, insurances, state business license fees, ITAR fees, etc. would run many tens of thousands of dollars and take quite some time. And, you will only be granted those licenses if you are going to be actually "in the business of" making guns for a living. Improving your own collection of toys is not a valid reason.