I'm sure @Grant will correct me if I am wrong about this but here are rules associated with names:
* All names you submit to contests are your intellectual property, not the CH's
* CHs do not have the right to claim any name in a contest as theirs unless they pay the creative for it
* SH encourages us to register our names
* If you submit one of your registered domains to a contest, you have to turn it over to the CH who chooses it as the winner, regardless of the prize in the contest and you cannot ask for more than that.
* Contestants can withdraw name ideas from any contest at any time, making it unavailable to be chosen
Therefore, you can register any of your names, any time you want, because they are your intellectual property and do not belong to any CH.
However, best practice would be that you would notify the CH that you registered the name (or, you can withdraw it and re-enter it as a registered domain, either way). You take a chance that the CH won't like that and will choose a name that is not already registered. OR, you can withdraw the name and therefore take it off the table as an option for the CH to choose as the winner.
SH would most likely "discourage" you from registering a name that is in the running for a win, simply because the CH might not like that you did it and SH wouldn't want an unhappy customer. But to my knowledge, there is no rule that says you cannot do it.
PS: I have never done this and you should Bluebutton before doing it if you have a question