Of course you can make a "cheap" 338 Lapua.
Anyone who has any knowledge of the 1917 Enfield knows they are quite accurate.
The 1917 action is a long action, longer the the Long actions of Remington, Winchester, & Savage.
1917 Enfield actions are relatively cheap. Either buying the action alone or boogered up sporterized 1917 Rifles.
A as Issued 1917 Enfield action is a different color, meaning they are going up in price because of the CMP's GSM (military surplus rifle) programs. But they have to be un-modified.
The 338 Lapua is based on a 416 Rigby case necked down to .338. The first rifle I ever made was a 416 Rigby on a 1917 action. The only difference between my 416 and a 338 is the barrel. So if you get a 1917 Enfield action, open up the bolt fact to fit the 338L/416 case, Open up the mag well allowing the thicker rounds to go into the magazine, stick on a 338 Lapua barrel, You have a 338 Lupa for the price of a barrel and action.
To make it look nice, you'll want to grind off the ears used to protect the rear sights, replace the bolt handle. Another cheap fix I did on mine was to replace the firing pin group with one that allows for cocking on opening instead of cocking on closing.
I built my 416 in the late 70s and the only problem I have had with it is I made it too light, it kicks like a mule.
I know things are different now, but in the 70s I was given the action, I paid $80 for the barrel, another $75 for a Bishop stock. $18 for the cock on opening firing pin group, and $15 to rent a 416 R reamer and headspace gages.
I did all the work myself and the total price was $188. Granted that was late '70 prices, but even at todays dollars, you can do it a lot cheaper then paying $4-5000.