Much of the advice given in this thread is useless as you are here in BC-Canada and sending a rifle to the USA is so much hassle that it is seldom worth doing, IMO.
So, you are confined to using Canadian smiths and good ones here are few and far between as compared with the States. Given, your relatively low budget, I would suggest a simple re-barrel of an existing Classic Mod. 70, especially a stainless version, or, a Ruger MKII.
There ARE some FINE barrel makers here, Ted Gsillard's sts tubes SHOOT and Bevan King, if still working, RKS, the same and a newer guy in AB whom I just spoke with and intend on ordering from soon, can all supply a good tube in .338" bore.
Smiths whom I can and do recommend, based on work done for me and on some pricey rifles, ie, my two Dakota 76-.338WMs and many others, are Jan Kolenbrander, VERY good, SLOW and a bit more costly than some, but, WORTH IT, Blackart, very busy, EXCELLENT work, good as Jan, super guy, maybe Bevan, maybe Nobby Uno, in Comox/Courtenay IF these guys are still working.
Others, whose work on friends rifles I respect, Denis Sorenson of the Victoria area and some young'un near Cranbrook,"Bill".... thinks he is "old", but, I am older than he is.....not as ornery, tho'....
IF, you need an action on which to build, look into the fine HVA actions now sold in used Husqvarna 1600 rifles by Tradex, I have two customs built on these and would do more if I were not so ancient and gimpy......
Lots of MK-Xs around and the fine, fine Brno 21s are out there if you look hard and these would be my first choice.
