I believe that the barrel could be machined for the small shank receiver. And PT&G makes a bolt head for the Lapua sized case.krttrkllr,
The .338 Norma Magnum cartridge is similar to the .338 Lapua except it's short enough to run through a standard length long action.
However, Savage uses the large shank action for this so your solution with the small shank and lack of Lapua bolt nose is not applicable.
From what I've been researching the biggest issue seems to be a concern for the strength on the receiver lugs.
"Yes Savage makes one themselves on the large shank (1.125 or 1 & 1/8"). That barrel shank size is large enough. However, the action itself isn't the same as the normal Savage Actions. It has a smaller ejection port for more stiffness, different heat treatment for added strength, as well as thicker locking lugs and thicker lug recesses for additional strength. Some have built their own on standard 110 actions but most will advise you that it isn't wise."
Savage also released a statement suggesting that not only was the small too small in diameter supported by most of the independent commercial barrel makers. Yes there is also mention of the smaller lug area to withstand the thrust. So basically you can go ahead with your project but at your own risk as it isn't supported by the firearms community.
You can get a stiller tac 338 action for about $1100 or get a complete savage 338 lapua for about the same price and have it rebarreled. If you go stiller you will need a stock for that action and bottom metal and trigger. Can get pricey. I rebarreled a savage to 338 norma and it shoots great.