Thanks for the tips. I realize this is a cheap kit, but it finished really nicely in flat black lacquer (like a piano finish) and sounds really good from those little potted humbuckers. I live near a Guitar Center store, where you can play anything they sell, and this kit sounds every bit as good as many of the real Les Pauls I've picked up there. Also, because I did the finish work and set it up with lots of time to do it, the fit and finish is actually better than Gibson's in some regards, and I like the flat, satin type finish, particularly on the neck. It doesn't drag and grab like a slick poly finish can.
I actually took this kit guitar to NYC after I finished it in November, where Les Paul still plays once a week at age 89. After the show I had him look over the guitar and sign it... right on the headstock, between the tuners! Les liked it, and when I described the hum problem to him, he said it was likely a ground issue. "Make sure you check the pots." That, coming from the man who invented the electric guitar! What a great night that was.
So, to make a long story a little shorter, I have a special affinity for this guitar and will spend a few bucks on it. I like these little humbuckers, so I don't want to replace them right now, unless they are absolutely the culprits (but they are potted, so they shouldn't be, right?). I am satisfied with everything but the hum.
Where would my money best be spent next? I am pretty certain the grounding issue has been put to rest. The Guitar plays sperfectly silent when on Toneworks and headphones. It sounds like I have a somewhat normal hum that could be reduced. Should I start by shielding and then grounding the cavities in copper foil? What about replacing any interioir wiring with shielded wire? Is that necessary, and if so, which wires would benefit?
By the way, I have a picture of Les discussing the guitar with me. Does anyone care to see it?
Thanks for the help here.