Hey, Shaun; Lots of questions (from me
-How much, in feet/meters do you need?
-Is the flat stock already primed or painted?
-Can you make a drawn profile from the end of an existing piece (as a template/sample)?
-Do you have a bit supplier reasonably close by, to take your scrap piece or template to, for matching?
For myself, I'd do that last item first and if that didn't work out I'd get a HSS or Carbide bit custom ground to an exact fit.
Chances are excellent that if it's a piece from the '60s or earlier, that it was done on a shaper not with a router...ie an exact match isn't gonna happen!
I'm with Charles on the 'let sandpaper be your friend' thing. You really only need to blend the actual miters, not the whole length, if it's really a close match.
*Nice illustration by the way!