I think you first want to use the wording change of mode to capture the major to minor change. You often read something like "...a change to the minor mode..."
Modulation is not a good choice, because the meaning of that word is very clear, and means only a change of tonic without any indication of what happens in the new tonic. I think the typical usage of modulate carries the expectation that new thematic material is presented in a larger form. You don't modulate a whole song from one tonic to another. A work modulates to a new tonic and introduces a new theme. That is the typical usage.
In contrast to transpose - changing the tonic, but not the mode, like Over the Rainbow in
A flat major transposed to
D major - the wording reinterpreted in minor may be a good choice. Over the Rainbow reinterpreted in
G sharp minor.
I think reinterpret captures the important point that choices need to be made about which tones get changed. Putting the music into the minor mode is not as simple as changing only the major third of a major key to a minor third. The sixth and seventh scale degrees are variable in minor, and while there is conventional handling of those two degrees it isn't a matter of hard and fast rules. Different choices could be made for different expressive reasons.
Translation seems like a good description too. When a write translates a work of literature from one language to another choices are made for one wording over another. Two translators could translate the same book differently. The same could happen translating a musical work from major to minor. Two musicians could translate the music differently.