Imagine this idea. Instead of mounting the fan housing at the level of the
ceiling tiles, mount it to a braced box (made for the purpose) at the level
of the floor joists. Same kind of box you'd use in a normal installation.
Try and choose the location so the center of the fan is where the center of
a ceiling tile would be. But, there won't be a tile. You'll carve up some
tiles to make a box - 4 tile walls going upward from the supporting track in
which the tiles would normally lie flat. At the top of the box, you'll
install a tile horizontally over the electrical box, just as if the tile
were a normal ceiling. The "bell" of the fan housing will hold the tile in
place. If the tiles have the same color on their edges as they do on their
face, they should look good where the verticals sit in the tracks. If not,
some painting effort should do the trick.
This plan would gain you a foot, or whatever is the distance between your
joists and your tiles. And, if you're installing a GOOD ceiling fan, like a
Casablanca, there'll be no shimmy in the unit to make the box shake noisily.
And, the box won't look bad. Plenty of large meeting rooms mount tiles in a
ripple effect for noise control, or to go around obstructions.