How wide is your continent (of the 21000-wide planetary map)? For transformations like this, you'd generally want to distort as little of the world as possible, so starting with a cropped continental area might be useful.
Annoyingly, it turns out that I didn't get GDAL installed on my machine again when I had to reinstall everything last year and it didn't go well when I tried again just now. In theory, it should be a matter of doing something similar to
https://blog.mastermaps.com/2014/01/...with-gdal.html (create a GeoTIFF using gdal_translate that describes the input image, then use gdalwarp to convert that to the desired output area). I can't tell you the exact command lines, but I think that the idea is to warp from one center in EPSG:4326 to a different center (you're basically doing an oblique equirectangular projection if I recall correctly).
Sorry I can't be quite as much help as I'd hoped to be. I did write a little windows version of the program I described above, but it doesn't properly recenter the image, just stretches it on a scanline basis as shown below. I adjusted the code in the post above to be what's used in this program because it had several errors in it.
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It also only works on 24-bit uncompressed windows bitmap files because the library that I was using doesn't handle images much above 8K without problems. In theory, you can read a small enough PNG or JPG with it, but it will only output BMPs.
http://www.fracterra.com/EquiSlide.zip is the executable if you'd like to try it. No guarantees about anything at all, as always (make a backup of your files before starting!) I really need to go about getting a security certificate for that site one of these years...