Well it appears you are treating your original map as being Equidistant Cylindrical and covering the whole globe and including the borders. Based on the dimensions of the map, the standard parallels of the projection are at 41.41 degrees.
So on your original map, the borders are actually covering up part of the map, everything between 41.41 N and 41.41 S is squashed horizontally, and everything outside that is stretched horizontally as drawn on the map. Unfortunately, it doesn't look like that, which is why it looks distorted on the globe with pinching at the poles and stretching at the equator. That's why those little peninsulas stretching up to the pole are pinched down to needles on the globe, and why they would still look so if you were to make a regional map in a suitable projection. If it's in equidistant cylindrical, then they really are needlelike.
On the other hand, your original map also has rhumb lines and a compass rose, which imply that it is in Normal Mercator projection (The only projection that preserves compass bearings globally). Treating it as such could actually fix things.
Treating it as Mercator means that the top and bottom edges would not be at the poles. You could shift it up and down a bit, but assuming the map is centred on the equator then after trimming off the borders, the top and bottom edges would be in the vicinity of 75 N/S
I've overlayed it on a Mercator map of Earth (Going to 82 degrees N/S) and reprojected into Equidistant Cylindrical (With standard parallel at 0)
You should be able to map this onto a sphere the same way you did with the original map.
Now with all that out of the way, if you decide to treat it as a Mercator map from 75 N to 75 S, then the continent in question would probably fit well in a conic projection. Your three main choices are Lambert Conformal Conic, Albers Equal Area Conic, and Equidistant Conic. The first can be thought of as preserving shapes best, The second obviously preserves areas, and the last is sort of a middle ground, and also preserves north-south distances.
For the version you have now where it covers a full 8th of the planet's surface, a conic could still be used, though with far more distortion. I might also consider Chamberlain Trimetric.
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