So I can stop giving myself a headache with all of this spherical geometry?

I can't say I was anywhere close to a solution yet, so that's good news.
edit: Actually, I should have thought of that before. You could do it even without Re:Map just using Fusion's Texture tool (though I'd hazard a guess that Re:Map probably has better filtering). I've rendered UV lens maps like that in the past; I'm not sure why it didn't occur to me to suggest it for you. Maybe because I was fixated on a Fusion-only solution.