Is it that much of a problem with thick carpet? Bare floors or just a throw rug I could see causing issues. Have you heard problems 24" up over thick carpet?
What I've done with my center is place it as far forward as the TV stand allows preventing reflections off the stand. I have it sitting in Mopads angled up to aim toward head height at my seating position. That's what works for me but I'm always open to learning more.
Either way there will be compromise.
Carpet will reduce the floor reflections but not eliminate them. I have personally found that near floor positions for speakers are less than satisfactory.
The next issue is that centers are usually MTM turned the wrong way. That maximizes vertical dispersion and minimizes horizontal dispersion.
However I understand vertical speaker orientation is not a practical option for most.
More, may be all manufacturers, should develop coaxial center speakers, which goes a long way to solving a lot of the problems of center channel speakers.
Anybody with construction skills should really consider building a center speaker based on the SEAS coaxial drivers. These drivers are grossly under utilized in center speakers.
They are the best currently available coaxial drivers and independent tests have found them superior to the KEF coaxial drivers. Properly loaded they are good neutral very low coloration drivers.
I seem to be the only one using them here and that is a pity. In my view they are simply the best drivers around for center channel use.
I have visited a lot of dealers now and have not heard a center speaker that comes anywhere close to mine on performance, especially stable uncolored natural speech.