Obviously the H&R/NEF handy rifles will probably be the most cost effective, but one could have just about any combo he wanted if it was built on a TC Encore frame...
Inexpensive was never one of The Colonel's criterion for the scout rifle.
The H&R Handirifle is certainly affordable.
The Rossi is even less expensive but it hit every branch falling out of the ugly tree.
The Encore is interesting but in my experience its ergonomics are poor; it just doesn't feel like a rifle. Don't lose the hinge pin...
The Contender carbine (at least before the G2) handles nicely but it doesn't come in 308, 7mm08, or 243.
The Ruger No.1 is fine starting point, but you'll want a custom barrel and a synthetic stock to make weight.
Triggertime's awesome recommendation for the Dakota Model 10 would make for a one of kind single shot scout. They'll put the barrel on it for you, but you'll still want a synthetic stock. Perhaps the factory knows someone who will make a hand laid kevlar and carbon fiber handle. $10k ought to get you started.
How about the Blaser K95 or the Merkel K3? They're light as a buggy whip out of the box (~5.5 pounds). The basic guns start at ~$6,000, which leave you plenty to send the barrel off to Grizzly Custom to have Lew silver solder some scout scope mounts to the barrel. http://www.grizzlycustom.com