correction! I was using alt-138 for è instead of 130 for é...really should have proof read before submitting.

After reading comments here for at least an hour, I had to give my interpretation of the use of the word 'resumè' in the framework of the English language.

Now, as we know, Rèsumè and Resume have the same spelling in English, in Australia I dare say that they are both pronounced basically the same (by the majority of people) up until the final letter, so therefore, for English users, I believe we should only use the inflection on the final 'e', to distinguish between both words when written.
Why must we spell it 'rèsumè' if we don't pronounce the entire word as someone from France? I believe most of us say 'reh-zu-may' for resumè and a toss up between ree-zyoom, or reh-zyoom for resume. All 3 words are constructed very similar at the beginning, which is why I believe we need not use the first acute E when spelling resumè.

It shows a common courtesy to the French language itself as we have lifted that word for our own use, especially as we still carry the inflection verbally on the final letter only.
I can't recall hearing any English speaking person pronounce both of the acute E's.

Another word that comes to mind which carries the same inflection in the English language that also holds its original French pronunciation is 'cafè'. The spelling of cafè still carries the acute E, as that sound doesn't exist in the English language unless we spell it to be so, like... CAFFAY or something similar.
It is also acceptable to not spell it with the acute E, as there is no other word that it can be confused with. The general English speaking population understand this and has accepted the English translation of simply 'cafe' without an acute E, we will still say it with the inflection, even though it is not spelled that way at all times.

My personal choice - resumè and cafè