If the integrated's internal amp's two channels can be bridged into one channel driven by one of the integrated's preamp channels, then yes, you can benefit from more power as a single channel amp here. I highly doubt the integrated has such a feature....but you never know. Maybe there are jumpers in the unit to provide this.
If the above is possible, you have one channel down, one to go. The problem now is to find another amp with exactly the same sonics as the newly bridged amp in the integrated. This is even less likely than the bridging option to begin with. The last thing you want to do is drive the speakers with "different" amps. And yes, the one IC to the external amp would certainly botch up the match between the two channels even if the two amps were the "same". And locking yourself into an amp that matches the one in the integrated is simply too limiting. This is a goofy proposition all the way.
I would consider any number of other possibilities before I went through the pain of what you have proposed:
1: If you want to upgrade a little at a time, find an amp that will drive your speakers to the musicality that you desire. I suspect that the integrated's preamp section will drive all but the lowest sensitive amps (typically matched with high output tube preamps) just fine.
2: Ultimately determine which is more important to you, the speaker or the integrated, and look to find a more suitable mate to the piece you decide to keep.
3: It looks like you are using a subwoofer. Is this not buying you a lot of headroom for the amp in the integrated to give the presentation more dynamics, control and slam? If you are driving the main speakers full range, perhaps you can crossover the speakers with a high pass filter as the sub is handling the bottom 2 octaves or so. This would free up a lot for the amp to more easily drive the the remaining frequencies in the main speakers.
4: Investigate other reasons why you do not have the dynamics and/or control in the sound that you anticipate is lacking now. As my sources have become more refined and resolving, I find that I play the music lower in volume.
Hope this gives you some ideas to consider.
John