Definitely put the Emotiova through the same test - your speakers at the same volumes and see how it fares!
Based on specifications, the 2200 is more capable:
Outlaw 2200
Emotiva A-300
Emotiva has superficially met the criteria of "doubling down" at 4 ohms, but at 20 times the distortion and only at 1 kHz bandwidth (as compared to 20Hz-20kHz)! Those are pretty substantial compromises to get the 300W rating@4 ohms, while the Outlaw is doing 300 Watts without having to give up THD or frequency range!
I consider this very poor form. Why doesn't Emotiva publish the power rating under more standard conditions for high quality gear?
That you are getting the 2200's to heat up that much makes me wonder about the load your B&W 683 S2 speakers present.
Here is the Impedance (solid) - Phase (dashed) graph:
I am far from qualified to interpret this graph, but the conditions between 600 to 700 Hz of <4 Ohms and Phase around 45 degrees seems to be pretty tough for any typical amp!
Hopefully someone with a better trained eye for these measurements can comment.
The Emotiva amp also disappoints me with how the two fans point straight into the cooling fins. In comparison to the QSC amp below, the fans in the A-300 look like they were added as an afterthought once they found out how quickly the unit overheated without a fan!:
Compare that to the QSC RMX-850 - their's has a tunnel with the cooling fins inside and the fan at the back draws air through the tunnel. This is so much more efficient!