New Bathroom Fan
I think that the easiest way would be to cut the round hole section completely out back to the middle of the nearest ceiling rafters on either side and straight across between the joists outside the hole with a sharp utility knife in order to install a new solid square or rectangular patch to work with.
A. Cut the existing hole out, keeping the edges of the remaining drywall as straight as possible at the middle of the joists.
B. Remove the old fan box and install the new fan box, connecting the wiring and vent pipe to the new fan box. Leave the box protruding down enough to be flush with the new 1/2" sheetrock patch, which is what you probably have on the ceiling...measure it to be sure. (See installation directions with new fan.)
C. Cut the new patch to fit the fan and the drywall that you removed.
D. Install 2X4s between the joists on the ends of the hole, half under the old sheetrock and half out to screw the new sheetrock to.
E. Screw the patch around the edges about every 6". Tape, mud and sand the seams to blend well with the rest of the ceiling. Prime and paint the patch and the whole bathroom ceiling, because I doubt that it will match without doing so.