ImageMagick can't do this—it doesn't do projection remapping. But there are a number of other tools that do. I'm pretty sure that Hugin or panoramas tools scripting or Gimp and Mathmap could get you there, but I'm lazy so I paid for a license for a commercial application called Pano2VR. Pano2VR can remap equirectangulars to cube faces. Because the cube faces are rectilinear projections, they'll be a closer approximation to what a 2D viewer displays than the equirectangular gives. You can then edit the cube face(s) in any editor (like the Gimp), replace it/them, and map all six faces back out to an equirectangular.

Another trick, if you're just trying to mess with the nadir (straight down view), is to re-orient the pano up 90° in tilt by dragging vertically in the preview, or setting the viewpoint explicitly (there's a numerical transform button in PTGui; not sure what it would be in Hugin), so that the middle of the pano is the floor and exhibits the least distortion. You then edit the equirectangular, and then readjust tilt back down to 90°. See John Houghton's Viewpoint Tutorial.

Addendum:

Nice choice of viewpoint. This is just my personal taste, but I'd also rotate the pano in yaw so you're in the center of the frame, not split at the edges, and fix the roll/tilt so your horizon isn't bowed and your verticals are vertical, like so:

This took about five seconds in Hugin's preview window's move/drag tab--dragging vertically changes pitch, dragging horizontally changes yaw, and right-dragging changes roll.

But, then, I'm persnickety about these things. :)

See also: How to reproject and crop a 360°x180° panorama?