None of the conversions I've seen have had any form of welding done, they are precisely threaded and locked with pins.
The original equipment barrel shank is used as a bushing, not a bushing made for the purpose. The shanks are cut from condemned shot out barrels.
The only reason these methods were used was because it was easier than setting up to thread an uncommon thread form or size, many European and Asian rifles use the Whitworth thread form. Most U S rifles use either a standard V thread or the square threads of the Springfield and Krag.
I'd sell that Mauser barrel and order a barrel threaded for the 1903.
I had thought you meant you wanted to use a cheap surplus milspec Mauser barrel. Small ring 7mm barrels sometimes show up at very low prices.