You write about your wife's state of mind, for which reason she wants to be examined by a doctor, while you do not care for it to be done so hastily.
While I agree with you that there is no need to be upset that she has yet to conceive, still, since your wife is upset you should explain to her your reasoning. If she accepts it in a good and glad manner then that is fine. If, however, for whatever reason she is still upset, you should comply with her desire to be examined by a doctor.
With regard to your hesitancy because the doctor is a male: I have yet to hear about being particular about this when it involves health matters. At most, you can investigate whether there is to be found a female doctor who specializes in this area.
Excerpt from Igros Kodesh, Vol. XIV, p. 100
It is my opinion that with regard to the specialty of gynecology, it is worthwhile
wherever possible that the physicians be women.
In addition to reasons of tznius, it leads to greater efficacy in choosing the most proper course of action. For it would seem that it is more natural for a woman - rather than a man - to more easily and completely understand and feel the nature and feelings of another woman (and conversely with a male patient).
Excerpt from a letter of the Rebbe, 23 Teves, 5748