I'm not trying to burst your bubble, just simple things to consider here:
1). The standard long action would not work for Edge length rounds, (.338 being only 2.5" case length), but if the .375 h&h being 2.85 case length/same with .300 RUM-parent case-....maybe...
Guess that's a good question
2). While the BAR is considered the "champaign automatic" vs the Rem etc. Your stuck with one of the worst triggers known to man, & no way to fix it. Not to mention a lot of guys I know that own them use small base dies & can't romp on thier loads too hard without causing "sticky" brass upon extraction. So ever comming close to reaching the edges full potential may be very difficult. I'm not familiar with exact p.s.i's of each cartrige here, but you may want to research it before hand so you could guess your potential.
3). I'd ask a gunsmith if opening the bolt face to accomodate the .404 parent case (may) cause your extractor to be weak.(maybe not). Plus, a barrel screwed into an aluminum reciver with no support (ie bedding/block of some kind) you might be stuck with a short sporter wt. barrel, again limmiting performance, would be another thing to consider talking to a gunsmith about. I'm not well versed in this arena.
I love my dads BAR in .338 win mag. It shoots like a dream, as do a couple of my uncles BAR's and one cousins BAR, but there doesn't seem to be much a guy can do to em to take it to the next level without a GREAT deal of hassle.
In My opinion only, enjoy them for what they are, as they are a joy to shoot in large magnum calibers/cartriges, but just not a good building platform. Again, this is just my opinion.