Great question... and to answer your question directly, military docs are not particularly known for being any more or less competent than civilians. In fact, the range of quality in physicians in general is perhaps more to blame for your bad experience than anything peculiar to the military itself. The only thing I can think of is that the meds for bipolarity will affect your performance in your duties more than a straight anti-depressant... but otherwise, there should really be no reason to pull a punch when it comes to diagnosis. The doc s strive for accuracy, despite possibly having a poor "bedside manner" (like you described in feeling like you weren't taken seriously). At the very least, poor communication on his part is clearly present.
But here's the deal: an anti-depressant will do one of two things, depending upon which "family" it belongs to... if you really ARE bipolar (and you have strong family evidence, although I - obviously - cannot make that determination/diagnosis over the internet) and you take a medication in the SNRI family, then it may kick off a manic episode, which would be bad. If you take an anti-depressant in the SSRI family, then your depressive symptoms may abate, but it will not reel in any manicky-symptoms, and again may shoot you up past "normal" (less likely to kick off mania than an SNRI, though).
So I would get another opinion if you feel that your symptoms are having a negative effect on your life, your military service, etc. Even if you are not full-bore bipolar, you may be either cyclothymic (sort of a minor-league bipolar) or have a condition called "Bipolar II" which means that you suffer through the depressive phases of bipolarity, but do not hit the top floor of mania - you only get to a point called "hypomania," which means that you have some of the sped up symptoms without being totally out of control manicky. If either of these is the case with you, the depression will be easier to spot/evaluate than the mania, so the bipolarity may be missed in favor of a depression diagnosis.
I hope that made sense! So get that second opinion before you proceed, if a second opinion is available. If you are satisfied with the response, please hit "Accept." That is the only way I can receive credit for my answer. Thanks-