The magnum long actions are beefier. I dont have scientific data on this, but I've got a 110ba and I've got a savage 10 and the 110 definitely feels beefier in all regards. Maybe someone will chime in and give you an official Yay or nay, but I doubt it.
I'm just wondering what your trying to win out on here? Savage 338 lapuas are about as cheap as you can get for the caliber. For example I'm selling my 110 for 1600 dollars right now in the for sale section. I can't imagine putting a better one together being much cheaper unless your skimping on something or getting certain things for free.... But the bulk of the cost would probably be the action and barrel anyways so.....
I would not skimp on your safety. There is some serious pressure that your lapua rifle has to take, unless your gonna be loading very light but doing this negates any advantage of the 338,seriously consider the 300 instead then.
If you want a 338 lapua on a budget, buy a 110ba and be done with it. Use the extra parts as backup or sell them off. You will be better of for it in the, reliability and safety department doing it that way, rather than sticking with what you already have and working backwards. Keep the barrel for when you shoot out yours (this is a magnum barrel burner remember), sell the chassis any any other parts to recoup some money towards the built rifle.
There's definitely pride in 'building your own', but when people do this it's either to save money or build something high quality and tailored to themselves. In a 338 there's no reason the save a couple dollars(you will blow more in ammo cost trying to find a good load for the Frankenstein) and if your doing it to build something high quality Savage is NOT the action to base it off for a 338. So the only reason would be because you have a spare action for some reason you would want to do a it's a fun project. But the action is the key, not the barrel/chassis etc...
If your dead set on it, I'm not sure of the hurdles or liability involved but an official gunsmith can probably order a 'replacement' action directly from Savage to 'fix' yours.
I'm not a gunsmith or professional anything in the shooting world for that matter, so this is just my 2 cents!