It turns out I manged to find a way to tonemap this monster of an image. Theoretically this should work for any image size up to 300000x300000 (90GP), which is the maximum supported by Photoshop. Of course, a full 360x180 panorama would end up as 300000x150000 (45GP). Either way, most likely not on anyone's radar, even mine. I suspect at those resolutions a dedicated panoramatic head is required, and its associated stitching software.

Onto the procedure!

1. Export from Autopano Giga in Radiance HDR format. I tried a few different options but in the end Radiance HDR provided the best compatibility in the next step. Time: ~12 hours
2. Using VIPS, transform this .hdr into a BigTIFF file. This is done using the vips tiffsave command, which takes HDR but not OpenEXR (the two 32-bit formats supported by Autopano). Time: ~15 minutes
3. Open the .tiff in Photoshop. Save as a .psb immediately. Time: ~1 hour

Overall, not a particularly involved process when you get it down. Unfortunately, it turns out that there was only one combination of likely hundreds of image processing programs and libraries, and file formats, which is capable of this. I expect to make more panoramas in this format, and most likely in the same size (using the same 135mm/3.5 prime lens), and so far this seems reliable enough on my system. I'll edit this answer later to show the completed panorama.