The OP did not provide information regarding the size of the room in which the amp and speakers will be installed, or about listening habits and preferences.
From a need for a solution that is no more than 7 inches high, can we deduce that space is limited ? This may be important due to sensitivity of the speakers. 88db in a large room with a modest tube amp may not be enough power, but in a small(er) room may be sufficient.
Many good suggestions were made. My suggestion is a bit out of the box: Go vintage.
There are several models of vintage integrated amps, that once correctly restored, will rival the sound of new amps, at a comparable (or lower) price. Vintage amps tend to hold value much better than new.
Sherwood 5ooo series amps are extremely well regarded, and sound wonderful. Sherwood embraced a slimline design, and all power tubes are installed at a nearly horizontal angle, minimizing the height of the unit.
Fisher X202 is an outstanding amplifier, but will take up more space than a Sherwood.
Fisher X1000 is one of the best integrated amps from the golden age. 40wpc of EL34 power and an excellent preamp section.
Sansui AU series, and Luxman SQ series should be considered.
One can find examples of all of the above for less than $1000 and then budget another $500-$1000 for careful restoration.
In all cases I would recommend inclusion of a quiet fan behind the amp due to possible ventilation compromises inherent in tight rack spaces.