Have either of these excellent trainers bothered to do any in hand work with your horse? I’m specifically thinking of proper rein back in hand - so full longeing tack including side reins, and the proper use of the butt of the in hand whip on the chest to encourage him to understand the voice cue? I’m betting not or that horse would be happy to rein back under a talented rider with a BNT trainer on the ground.
As for a pulley rein? Was the horse about to bolt into traffic or some other dangerous situation, or in a continued out of control gallop? No? Then, no to using a pulley rein. It’s an emergency brake only.
Some horses can be stubborn about learning rein back - it’s not entirely natural, especially after we’ve spent so long instilling FORWARDS!!! and it can be uncomfortable for those that are weak(ish) in the stifle area. But, just as teaching turn on the forehand from the ground makes teaching it under saddle so much easier, so does teaching rein back properly from the ground. Physical cue tied to voice cue = far greater understanding when you’re up top and ask for it. I think often times, as dressage riders we’re so afraid to get into the habit of using our voice that we forget what a powerful aid it can be when used well.