I'm trying to replace the ceramic tile in our bathroom. The problem is
that there is a gap around the toilet drain pipe. The subfloor is not
quite flush w/ the drain pipe, and there is an even larger 4" x 2" gap
in the concrete backerboard that the tile rest on (the previous
handyman just filled in the gap w/ putty).

My question is: can can i best fill this gap? my concern is that even
if i replace a portion of the backerboard, it won't be supported and
the tile or the grout between them will crack. The back corner of the
toilet rests in this gap, so there will be pressure on it. Right now I
have a shim screwed between the floorboards to support the toilet, but
i'd like to have something more durable than a shim. Any ideas?