Where the islands end up will depend on the projection used. I used Mollweide for that particular example as it has evenly-spaced straight lines of latitude, unlike Mercator, which has a hyperbolic tangent spacing between lines of latitude. On a Mercator projection they would end up on the same side of the globe, but be greatly reduced in size. Your original map showed equally-spaced lines of latitude and I picked Mollweide as an example of a projection without worrying too much about final distortion. Mercator will keep the shape of the coastlines, but will radically distort the size of polar regions; the Mollweide projection keeps the latitude scale, but starts to distort the shape as it heads poleward.
Most folks draw a map with the goal of making it look good and later try to shoehorn it into some sort of "realistic" projection. The essence of mapmaking is to produce a work of art that is suited for a particular purpose and that balanced the technical and visual aspects. Given a choice, most folks will fudge the technical to get the visual. Mountains, for example, may be represented as groups of small pictures, or as chains of hachures, or some other way entirely. None of these are technically accurate representations, but they convey the sense and general location of mountains, which is often an important part of the goal of the map. Such a representation for mountains is most likely totally unsuitable for map used to plan the route of a road or canal; those purposes require a different sort of map, usually one carefully surveyed and reference to ground truth with a precise projection.
The reason I mention that is that if you're concerned about technical accuracy, you're probably better off deciding where you want things and how you'd like them to relate and then redrawing your map to take into account your understanding of the way you want them to be against your understanding of the overall globe. Early maps are often inaccurate representations of reality because accurate longitude surveying requires good timekeeping. Changing the map a little to match a newfound understanding won't change the underlying reality of the world (such as it is) and inaccuracies in early map editions can be explained as the work of some not-so-good surveyors that were further distorted by someone who wanted to make certain areas more important.
ReprojectImage assumes a spherical world, which differs from earth by about 1 part in 300 (pretty much close enough for any reasonable visual work). It doesn't know or care about a world size because the math here is all about the angles, not the actual distances.