I used to play bombarde in a dance band, with two conga players. I played oboe, too (not in the band), but that's different. I held the bombarde reed entirely in my mouth, and held the oboe reed just slightly inside with my lips on it to restrain and control it. The bombarde seemed at its best when unrestrained and just gloriously loud and abrasive.

The reed has to be very wet. That makes it easier to play and less ticklish. Soak it for a long time before playing, and keep salivating on it till you're done playing. Without the saliva, playing dries it out.

I don't know of any way to mute a bombarde, but it might tone it down a bit if you use an oboe reed or one of those plastic practice reeds for a bagpipe chanter. And it might seem a little quieter if you play it near the exhaust of a jet engine.