It seems to me that you have a more fundamental problem here: what happens, in your competitive play-by-post RPG, if some people post more than other people?
What if one group has finished all their actions for Monday and Tuesday and a different group is still working through Monday?
In other words: how do you measure time?
We mostly play pencil-and-paper RPGs here; to the extent that we measure time, it's mostly "time only elapses when traveling". Combat and conversation are so fast compared to travel time that it doesn't make sense to count the time cost for them. (Occasionally people will attempt something clowny, like searching an entire building complex for secret doors, and we'll have to assign a time cost to that too.)
Once you figure out how you're measuring time, I predict that your solution to sleep time will just fall out of that.
Here's an example. You might imagine a game where the game master says: "During this week, August 20-24, we're playing out the actions taken for Day Three of game time. During this week, you guys have twelve time units to spend. The following actions will consume time units: (traveling, searching, crafting, resting, etc). If you spend all your time units early, you can still take "quick" actions like talking to people, roleplaying, fighting things, or making knowledge checks, but otherwise you have to wait for next week, when we'll start Day Four and you'll get more time units to play with."
There are many existing examples of online RPGs with models like this. The one I'm most familiar with is Kingdom Of Loathing, in which you get a number of "adventures" per day which resets at midnight server time. These RPGs generally don't have game masters, but it seems to me that something very similar would work for play-by-post.
The weekend could be spent on nighttime activities, or it could just be a break from the game.
