Forager, I have done a few barrels for the exact same reason you're stating.
After many failed attempts in an effort to get this stuff out, using chemicals I shouldn't have, I settled on "Fire Lapping" with JB Bore Paste (Compound)
Saturating a patch, that I could easily start by hand, not a thick patch, but maybe a little thinner by just a few thousandths under the size that you normally use, then after a dozen or so shots, clean it, dry it well, and check for that dreaded ugly brown.
If it's there, do it all over again....that should do it!
My son in New Mexico did his barrels, all of them, with Valve Grinding Compound first, then followed up with a half-dozen Fire Lapping shots using JB Bore Paste. He told me those barrels had never been so clean, and he never got the brown anymore!
I haven't talked to him in years about this as we both just considered it "a given", but I will ask him, once I remember it.
Mine are all fine, along with other folk's barrels that I have helped them with so there has been no reason to think his barrels would be any different.
Something you may find interesting;
Once you do this, and I have heard several say this same thing, you can use Bore Butter again, just not regularly, and not get that dreaded ugly brown patch back......all this tells me, and it's just a theory among myself and a few others, that we, collectively, just don't do a good enough job cleaning/scrubbing that barrel when it is brand new, before we shoot it the very first time.
As for myself, I will still use Bore Butter on occasion, and after a good cleaning, it is never as bad again as it once was many years ago. Sometimes a little stubborn maybe, but it comes out with just a little extra effort.
Now, I must say that there is a group that believes you can do the same thing, and get the exact same results, by Hand Lapping with a saturated bore Mop. I have no problem whatsoever with that.
There is also a group that believes strongly that you can "season" a bore using Bore Butter. I have met and talked to a few and I would never condemn, or even try to change their thinking.....but, who knows? Perhaps they are innocently confusing brown patches with a seasoned bore.