My understanding is that the advice about not reversing a bore brush is more to protect the brush, than the bore. The bristles on a bronze brush tend to break off when you do that. I can't see how the much softer bronze can damage a steel bore.
Still, I generally follow the advice, and don't reverse the brush.
I find that a nylon-bristle bottle brush works well on shotgun chambers. You can really scrub with no danger of damage to the chamber. True, nylon is not as aggressive as either a bronze bore brush or a proper chamber brush. But I find that with regular attention, the combination of cleaning the chamber and bore with the bronze bore brush, and a little scrubbing with the nylon bottle brush and solvent, keeps the chamber very clean.