Bobby Hoyt in Fairfield pa can do nearly anything barrel related.
Relining is an option, but if I were wanting to push conicals beyond 100 yds, the higher pressures associated with conicals may be too much for a "lining". If the liner swells even a little (a thousandths of an inch,) you have a ringed bore
The labor and work and cost of the liner are probably a lot more than the cost of a good barrel blank of proper width with the twist you want. Breech plugs compatible with the Lyman tang are available from TOW and others. (As I recall, TC plugs work with very little fitting. Then it is just an underrib and barrel tenon installation. Plus, you have a stronger barrel than one that has been lined.
I had Bobby re bore a few guns years ago. His cut round ball rifling was outstanding.
I have an Investarms Italian Hawken stock, to which I fitted multiple assorted parts from Lyman, TC, etc. I used TC breech plugs for the barrels I fitted and the rest was mostly cutting dovetails for the wedge tenon and sights. I used "silver bearing solder" (lower melting temp, but still sticks well) to fasten the undrribs and ramrod thimbles. (Over a two burner gas Coleman stove so I could heat that length of rib uniformly) They are holding up 25 to 30 years of hard use later.