I have been shooting Barnes bullets (both varmint bullets and the big-game TSX and TTSX) in over a dozen rifles for many years. I also shot a lot of other brands of the copper bullets. There could be some truth about the differences in the copper in the bullets causing bad behavior, though I have not encountered it - but I also don't let much copper build up in my bores. I have tested loads, alternating between Barnes and leaded bullets with no issues seen affecting accuracy when shooting the Barnes again - so I'd tend to think that should not be much of a concern, if at all.
One thing I found to be universally true with all the loads I tried in all the calibers - and that is that copper bullets are much more "powder picky" than lead core bullets. Because I have tested the Barnes with not only my rifles, but have dialed in loads for many of my buddy's rifles in the same calibers, patterns appeared with powders that shot well in multiple rifles of the same caliber with the Barnes. For example, 7 MM Rem Mag and Reloader 22, which shot the best of EIGHT powders I tried to get my first 7 Mag to shoot. Before that powder, I couldn't get better than about 3/4" - with some powders shooting 3" MOA. The Rel. 22 shot 120, 140 and 150 grain TTSX into roughly 3/8 MOA. It did similarly in three other rifles. Seeing how powder finicky these bullets are, I really don't know how accurate I can get them. I just stop once I get to under 1/2 MOA, except for my varmint rigs where I want 1/3" MOA (using the Varmint Grenades).
There are many variations in loading technique to attain accuracy with a load. some focus on seating depth first, others focus on powder first, etc. I can tell you with great confidence that the typical Barnes bullet will favor ONE powder far above all others and you may have to try several to find it. With lead-core bullet loads, I often wrestle with which of two (or three!) great loads shot better. That is seldom the case with the copper bullets. More often than not - a lot of powders shoot "O.K.", if not poor. But there will one (two if real lucky) that will shoot well inside that 1/2" MOA that I seek. These bullets do like to "jump", so yes - seat them rather deep - and I'd suggest using Barnes seating depths to start. Barnes, I think, did far more testing than most bullet maker manuals on various rifles for accuracy before recommending powders. They typically don't list the "20" or so powders many manuals will. They only list several which apparently were the best in most rifles. And it is refreshing that usually, one of those few powders will work the best.
What caliber(s) are you working with?