I will second Bramble's notes of caution to whomever may end up viewing this thread in the future.
One's ability to possess a TIG welding machine, and to puddle a bit of parent material with it's arc does not so simply qualify them to assess the possibility or feasibility of a weldment.
I have seen numerous 'expert demonstrations' of TIG welding in recent on-line video format which were ultimate hack-jobs, and can imagine that there is a much greater mis-understanding of the process than I would guess.

There is much to consider around such things, especially when such things are to eventually be parked near the temples of a hunter and run time and again on into the generations.

Always err to the side of caution, and with respect to the nature of the materials.

Again, to refer to the WR seen at the top of this thread - to take it as an example for the original proposition instead of some H&H -- I'd take a modern-built WR drop lock rifle made of modern materials before I'd take it -- in a hot second.

Chances are that if I were to order one, I'd order it in a similar configuration (save for the sleeve-job), shoe lumps and all!
And again, that is if the old and new were served to me on a plate, the choice of the two being mine as gift...

Cheers
Tinker