The following ought to keep the weight and price to a minimum, although the fantasy wheel build is $415 in components alone and could be cut in half by gaining a pound or two.
Take a Trek from '83 or earlier - the higher-end, the better, for more "planing" although a 6xx is just fine. Squeeze the chainstays and rake out the fork to get about 35mm trail. Install 650b wheels (Pacenti Brevet rims, one of those fatty White Industries rear hubs, and an SP SV-9 front hub) with Pari-Moto 650x42 tires under old Lefol or new Honjo fenders. Suntour Cyclone M2 GT or XTR M952 rear mech. Suntour Superbe front, or no front if you're one-by-curious. Bar-cons. Your favorite brake levers, cranks, bars, saddle, and pedals, because anything will do and these are personal choices. American Classic post if you want setback, Thompson if not. Mafac Tandem or Raid brakes. Brazed on or bolted on, your choice how intense you want to get. Decaleur and front rack by Gugie or Nitto. Lighting by Busch/Muller: Cyo Premium front and Toplight Mini rear.
Sleuth out deals on these and hey presto, you have a '50s style randonneuse that will eat gravel for lunch, for under a grand.
Credit where credit is due, I am basically describing
lonesomesteve's Frek.