Constant Peak Times

[Atoli]

To answer your initial question:
No, there is no server that is evenly populated at all times. Every server (even including Balmung) is on a certain datacenter and therefore has a certain timeframe attached during which it is evening for the overwhelming majority of the players on the server, and outside of that timeframe, there WILL be less players online; depending on the size of the server "less" may mean anything between "none at all" and "still quite a lot".
If you do not want to alternate between servers, you will have to deal with that.

@your timezone: So I assume you are East Coast NA?
When it is 2-5 am where you live, it is 8-11 am in Europe, which means most people are at work/school already, so European Servers (Cerberus, Lich, Moogle, Odin, Phoenix, Ragnarok, Shiva and Zodiark) are not gonna be of much help.
Timewise, a JP server would be best, since it is 5-8 pm in Japan when it is 2-5 am at your place. Keep in mind though that the English-speaking community is barely existent on JP servers, so that may make it unplayable as well, even if the peak times match up with your playtimes.

Honestly, I'd say just find a very populated NA server and stay there; while there will be less people active at night, it won't die out completely like on mid-size and smaller servers