I decided to go ahead and experiment with the crossover, since there doesn't seem to be much data available.
I'm using a miniDSP 2x4HD for the mid-high crossover, and it has a variety of options available for the filtering. I originally set it up so that I have a 1 KHz crossover, both filters 6 dB/octave Butterworth and listened to it for an extended period of time to be sure I had everything else right. The miniDSP has a nice feature where you can set up multiple configurations and switch between them, which was very helpful in that I can make two configs with only one parameter changed between them, and switch quickly while playing music.
I set up a second config with a 12 dB/octave Linkwitz-Riley (which looked like the best option among the higher-order crossovers) and fiddled with it for a while. I did have to adjust the mid/high relative levels (and of course the tweeter phase, it being a property of the 12 dB L-R that it's reversed) because there was less driver overlap and it affected the mids a bit (this was evident by listening alone, and confirmed with sweep measurements). Once I had this config set up to my liking, I did an extended session where I switched between them and tried to hear differences.
The result? For all practical purposes, I can't hear a difference that I can put into words. I think maybe the 12 dB crossover might be a little better through the midrange, though it's anything but dramatic and it might be a result of the level adjustment as much as reducing the driver overlap. I spent a lot of time listening to selections which demanded good transient response because that is supposed to be a potential weakness of the higher-order crossovers. Again, nothing substantive to report.
The main difference, which is not imaginary, is that my tweeter amp is not being asked to deliver as much power now; the power meter indicators are clearly not registering the same values as before. This is not a surprise, as it's one of the benefits I hoped to accomplish by switching crossover types - the amp is being asked to put out much less power below 1 KHz. I don't see an obvious difference on the mid amp, though surely it's not being pushed as hard above 1 KHz now either. These are good things.
So, preliminary results seem to be positive. The system still sounds great and possibly has improved a minuscule amount. I've learned more about integrating crossovers and though the information is mostly anecdotal, I wanted to share it anyway.