For your amusement, here's what happened when I used MMPS (under Cygwin) to project the south pole in gnomonic (where one could make appropriate changes, but I didn't)
south_pole.jpg
and here it is reprojected to equirectangular. One could then use appropriate tools to merge it with the original artwork, although MMPS could do that for you, too.
eq_south_pole.jpg
Here's the script that I used.
Code:
#! /usr/bin/sh -x
#
# project equirectangular to gnomonic
#
convert detail_colored.jpg tmp.ppm
project \
gnomonic \
-scale 2 -lat -90 -long 0 -w 2500 -h 2500 \
-f tmp.ppm >tmp2.ppm
convert tmp2.ppm south_pole.jpg
#
# project gnomonic to equirectangular
#
project \
-i gnomonic \
-scale 2 -lat -90 -long 0 -w 5000 -h 2500 \
-f tmp2.ppm >tmp3.ppm
convert tmp3.ppm eq_south_pole.jpg
"project" is the MMPS re-projection utility and "convert" is the ImageMagick image format conversion utility.
p.s.
Note that .ppm is a lossless, uncompressed image format. Of course, any intermediate editing should be done in a similarly lossless image format like png or bmp, for example, using a lossy format like jpg only for the final image.