A slightly different spin on FrenchMajor's point -

some people may be indifferent/unaffected by the choice of accents here, subscribing to the 'they are all equally valid' camp.

Other people may have a strong preference for one choice, or a strong dispreference for a particular variant - someone may consider two accents pretentious, or one accent ignorant, or no accents confusing.

Personally, I do not worry about confusing anyone by using no accent. If they cannot understand from context, then misunderstanding that word is probably the least of their problems in understanding my prose; and in written form I don't care about helping people pronounce English words. Having previously used the one accent version, it now appears obviously wrong to me, and I wouldn't want to appear ignorant to someone who in other ways shares my linguistic preferences; conversely, I have no desire to appeal to anyone who has a very strong preference for the single accent version. FWIW, I will probably use the non-accented version in future, given resume has passed into English in the same way as words like cafe that no longer require an accent; unless the intended audience gives me reason to adopt the double accented version.

There is literally no etymological justification for American Heritage's choice of the single accented version, particularly given that they note that it comes from a verb with an accented first 'e'. I can only imagine they adopt that to help with phonetics, as (in my experience anyway, though others may disagree) the common pronunciation of the word places an acute accent on the last e but not the first. But adding accents is not a characteristic of English (or American English), regardless of convenience. If you wish to note that it is a word imported from French, accent twice; otherwise don't accent. To take the half-assed approach is to create your own rules of language rather than following those already in existence.

For what it's worth - MS Word does NOT like the single accented version... which is why I went looking on the internet for answers, as I do not like to defer to Bill Gates without just cause.