Not that easy as you'd need some way to attach the foamer nozzle of the foamgun to the BHB. It won't just thread into the hole and if you round the foamgun nozzle and thread it IMO you'll compromise the foamer's functionality.
I've given this a *lot* of thought and it'd be quite a project. And even then you'd compromise the idea to some extent. When you move the foamgun up and down along the bristles, they all get lots of rinsing/flushing/etc. If you fix the foamgun's nozzle in place the suds will only be directed at that spot (not along the whole length of the brush). So from the standpoint of functional effectiveness, it could be that the high-effort, big-hassle manual method is gonna be the best.
Oh, and be careful with that (Braun, IIRC) BHB. Some are fine, others are coarse and will instill micro-marring.
Before going with the foamgun, I experimented with using a siphon-feed (made for applying liquid fertilizers through a garden hose) to add shampoo to the water. It didn't work out well for me, but maybe somebody else will have better luck with the idea. I was trying to do exactly what's under discussion here: pump soapy solution through that flow-through BHB. Even when it did work, it wasn't anywhere near as good as using the BHB in one hand and the foamgun in the other, but that was just *my* experience.
