CBGs normally have a much thinner neck than a 6-string and between two to four strings, though three is more typical. The neck normally extends all the way to the end of the body. So with an LP neck, you'd really need to trim it down considerably and add an extension to the tenon (if it has one and isn't a bolt-on type). With the trussrod, especially if you don't cut it down considerably, its likely to be very neck heavy

It will probably be easiest to get a 50 x 25 mm/2"x 1" length, and at least 914mm/36" long, of something like maple (or any availble solid hardwood), and make your own neck. No truss rod, no separate fingerboard, keep it simple. Three strings (typically GBD for open G), the top 3 plain strings on a guitar. Knock some frets in yourself and away you go. Lots of guides on the web.