for this job, it's completely irrelevant. you are not trying to run a local company and command over 100's of local men. you will be interacting almost exclusively with other Americans. The only interaction with locals will be with the service sector (airport personnel, taxis, food, hotels, etc) and people doing those jobs are trained and deal with foreigners same or better (female or not) than in the US. Actually, most of them are not locals either, but foreigners themselves from other countries. So, you would have to try really hard to actually see a local, much less have any concern about interaction. You may have to change your dress a bit, but that's easy.
and at the odd chance you will come in contact with a local man, many of them have different expectations of foreign women vs local women and wouldn't expect you to behave like his wife or sister.

Frankly, as a solo female traveler (on business or pleasure) i have harder time in the Caribbean than in Middle East due to the hassle factor from the men, lack of service and just general bad attitude in many places (or unwillingness to cooperate/do work). In Middle East, Ive' been always respectfully treated, with good service and willingness to communicate to resolve issues. to be clear, my experience involves gulf countries, levant (jordan, etc) and north africa (egypt, morroco, etc). Afganistan, Iraq and other war zones are obviously on a different level of difficulty. but at that point, i doubt being man or woman makes your life significantly easier or harder..