Your question startled me and I needed a couple days to let the ideas float a bit. I was initially stumped because I couldn't think of example of Rhesus or chimps performing similar acts. It could be that transvestism in a cultural "spandrel" ... that is, it has no direct evolutionary or adaptive function; it contributes nothing to fitness, to producing the next generation. It might, however be founded on psychological adaptations that have and had other functions. Some possibilities are:
1) There has been extensive discussion of the mating tactics that derive from the relative scarcity of eggs (females) and abundance of sperm suppliers. Estimates are that in some creatures 90% or more of males do not mate while nearly all the females do. By using female tactics, a "non-competitive" male could attract a wider range of social or sexual partners even if not in traditional roles for the production of children.
2) Female garb in our cultures is usually more festive than that for males; this tendency is opposite that seen in many other species where the male is the glamourous one. (We do have "power suits and ties" wherewith males display their relative social power.) Transvestites might gain social dominance within a small group through innovate clothing; a standing they may not attain with traditional garb.
3) Male behavior is female driven. The whole issue of male competition, especially in terms of size or strength, is thought to be a reflection of female preferences. Taking on female clothing might be a tactic to give a male significant influence over other males. "Influence" includes such things as being in the spotlight, attempting to "fool" other males, or getting social attention that would otherwise be missed.
4) Finally, our brains appear to be sexually neutral near conception and "sexually tuned" during gestation. Thus, all of us display masculine and feminine traits at varied intensities. Thus, there is a frequency distribution of male or female traits in people with female genitalia (which themselves vary in structure) and of male or female traits in those with male genitalia. Thus, most of us have the foundations for male or female behaviors; however, those adaptations are not expressed without many other events also occurring.
Good luck!
PS: Thanks for the reference to "Evolutionary Psychiatry" although I hate the title. What's next? Evolutionary Social Work? Sociosociobiology? Or Clinical Social Ethology? Evolutionary nursing? Certainly, Evolutionary Principles for Nursing Practice.