Having gone down similar rabbit holes quite a few times myself, I would now suggest picking up an older (pre micro groove) .35 Marlin or you could likely even find an 1886 Winchester in .33 Winchester. When you consider the costs of a new barrel or a rebore plus a reamer and likely paying someone to do some of the work, a used '86 is occasionally still found for reasonable money in .33 and I actually prefer them to the .348 in a model 71. Any '86 is a way nicer rifle than a '94 in my opinion. A used .356 in a '94 or a .35 in a Marlin are the easiest routes and no deer or black bear will ever know the difference. With the price of any decent factory rounds, every caliber requires reloading if you want to shoot any amount at all so its really down to component availability.
For .338 hunting bullets, the Hornady FXT is available and Woodliegh also makes an excellent jacketed bullet for the .33 Winchester if you can find them. Quite a few people have also reported positively on filing a flat point on a 210 grain, .338 Nosler Partition for use in the .33 Winchester. Lastly, the old 200 grain Hornady FP bullets are still to be found occasionally. I just box a box since Christmas from someone on here. There are options if you are set on the .338 caliber.
On the other hand, your idea sounds like a cool project so keep us posted if you go that route. A 200ish grain, .338 caliber cast bullet performs very well but is not a significant jump from the .32 special's 170 grain bullet which everyone knows is much more powerful than the .30-30! There I go flirting with logic and sentimental reasoning again...