@ Pdaines " From a linguistic perspective, resumé seems the most rational. Résumé would imply the French pronunciation ray-zu-may, which is clearly incorrect as well as awkward"

Actually it is not. You are assuming that the French pronounce the last é like an American. It would sound stupid to apply the same sound to both e's the way we say it: rAy-zu-mAy. However, when broken down, most French native speakers would pronounce the é as reyh-zu-meyh...with less of an emphasis on the "Ay" sound. It sounds better when you say it like that, and not awkward at all: reyh-zu-meyh

My personal taste is that we compromise and spell it the way we say it as Americans, which is "resumé". We pronounce the initial 'e' with an eh sound, not 'ay', but we do pronounce the second 'e' with an "ay" sound; the spelling of "resumé" reflects the American pronunciation of this french word. Personally I HATE when café is spelled cafe because my mind can't help but turn the pronunciation into something that sounds like "kayf"

source: my entire family speaks French and my mother's native language is French