Not that I have heard much of the Bryston gear, but considering your speakers and their moderate efficiency rating and 4 OHM rating you are getting 500 watts per channel as is.
This should, based on the math from 1M away from the speakers be netting you 116DBS; 2 DBS over what the speaker manufacturer claims is the peak volume output from the speaker.
Doubling the power by adding another amplifier will get you more head room, but I think you are supplying the speakers with enough juice with just one amplifier as is.
Unless the crossover presents a tough load on an amplifier, I don't think anyone could say for sure you would hear much of a difference.
Perhaps you would get more driver control, and thus tighter more articulate bass; but this is assuming that your current amplifier in and of itself is having trouble.
Perhaps, using that 4K+ that you would spend on a second amplifier, in other areas (e.g. power conditioning, PCs, interconnects, electronic upgrades, etc, etc), would yeild a greater overall sonic improvement.
In any event though, more amplification is never a BAD idea; but is it the most efficient way to spend money for a large sonic gain?
