I can't figure out how to add a comment so I will put this in the solution and let the moderators scramble to figure out why I can't comment.
My first impression when reading your question was "Why are you not designing your circles in a conformal projection like Mercator". You could project this map into a Mercator projection and see your circle and text distortion, fix everything to look nice and when you project it to your globe, the shapes should stay correct (that is the definition of a conformal projection).