Your situation sounds pretty much like being married to the wrong woman
I took some time to study each of your posts as I'm also shooting a 338 RUM which started life as a 338 Win Mag. The WinMag bbl was replaced w/a REM new take off sporter barrel - 26". It has a Holland QD brake installed along with a Limbsaver recoil pad on the other end.
I Tubbs'd the "new" barrel as part of the break in process.
My first input would be about your statement: "as soon as my shoulder heals up". My rifle weighs in at 10# all up and I'm attempting to shoot the barrel out to get a halfway decent excuse to upgrade to a bigger cartridge.
I shoot it prone all afternoon and have no recovery time. What kind of brake are you using??
Next, I can't get anywhere near the velocities you are getting. An inch+ advantage that your barrel has doesn't provide "that" much more velocity.
Considering the excessive copper fouling even after the Tubb's process is something to be considered.
What if the extra velocity, if it is really extra, and excessive copper fouling would it be silly consider a loose bore? What that means I have no idea but it may also make a little sense that it doesn't like boat tails. It's my under standing that flat base bullets and RBBTs seal at the base a bit better at the base upon firing.
If all else fails one last thing to try, which is a bit selfish on my part as you are posting some good stuff.
Have a go with US-869 and 300gr SMKs. Hodgon's says 101 grains is max but I'd look it up first. I got nothing to shoot in my 338 RUM until I got to this combination. Now she's 1/4 MOA @ 200. 2730 +/- 1 fps @ 40* F.
PS: with a Lilja which are tight bores, your pressures will go up and velocity will go down but most certainly the accuracy will be there if all else is in order.