I am not too sure of what a medium weight barrel is but......seems to vary with folks I know.
There used to be Redfield two piece mounts made up for the No.1B factory taper(Shilen #4 is right close) that show up every now and then on the various internet auction sites. These were for the turn-in fronts and windage rears. Discontinued some time ago, but old/new stock is still around some.
Talley make make a set of gunsmith basic bases. I have used some that were for Ruger rings, ie Ruger dished recoil lug machined in with flat bottoms. Pretty tallish, as to allow for custom fitting to whatever barrel. I used a set on a custom martini once.
The good thing if one calls it that; is that the Talley and Ruger bases are 1/2" nominal in width. Saves some weight and I think look nice and slim on the barrel. Weaver types are nominally 3/4" wide.
As you well know, always are custom cut bases; for Talley or for Ruger is not a big job on top, but the front base bottom curves are fairly complex.
Guess one could peruse the Talley or Ruger base folks for a tallish standard type base that could be touched on the bottom for said new barrel.
Final thought is to buy a straight tapered barrel and machine a sight base ala pre64 model 70, the same diameter as the breech. Probably need about 3 inches length for scope rings, and if solid, adds weight. The tapering on a lathe is not much fun but would ease the mount requirements. I reckon one could machine up a bushing to be silver soldered as the front base's support/interface on a more standard tapered barrel.