JME with Barnes bullets in 7mm-08, 30-06, 300 WM and 375 H&H, but my rifles always shoot them better (more accurately) after they are fouled a few shots with cheap cup and core ammo. Seems like copper fouling is copper fouling whether monolithic, or cup and core, but my experience hasn't shown they foul the same. So, I would suggest you shoot cup and cores first, then the Barnes. May not hurt to try that and see if your rifle performs similarly.
Just a thought. Best of luck finding what factory ammo works best!