If you try to use anything that attacks copper fouling, it will go after the copper in the bronze brush bristles as well, so you don't want to use a copper-removing cleaner on them. You can soak the brush in anything that penetrates well. Afterward, I expect an ultrasonic cleaner would blast off much of the loose fouling. However, as the others suggest, you'd have to go through a lot of bore brushes to pay for that.
My personal approach has been to get away from using bore brushes altogether. Most folks don't let solvents work long enough before patching them out. Common solvents can keep working up to 20 minutes or so. Gunzilla can deep soften carbon for weeks if the carbon is deep enough and hard enough to require it. Letting Gunzilla sit overnight has let me patch out carbon rings at the ends of chambers that even a brush won't touch.
Note that carbon hardens over time, so anything you can do to wet it with solvent before you leave the range will make it easier to get out when you get the gun home. I got some Wally World travel-size (within the 3 oz TSA limit) pump sprayers and put Bore Tech Eliminator in them. At the end of a range session, I pull the bolt and pump a few squirts into a rifle bore–enough to see it run down to the muzzle. I use their Rimfire Blend after shooting lubricated lead bullets. Then I plug the bore of a rifle at both ends or put the handguns in big Ziplock bags and into their cases and drive home (just under an hour). Pretty much everything patches when I get there. Reapplication is needed with stubborn rough bores, but the 20-minute rule seems to work pretty well to get them clear.
Tough job carbon gets Slip 2000 Carbon Killer or an overnight soak with Gunzilla.
Tough job lead fouling gets Sharpshoot'r NO-LEAD
Tough job copper fouling gets a dose of KG-12 followed by Cu++ to check for signs of remaining copper.
But with the exceptions of an old gun crusted hard with carbon fouling or using an undersized bore brush as a wrapped patch holder in a blind breech semi-auto for cleaning from the muzzle, I rarely use a brush anymore.