I use the rubber stopper setup too. Make sure the beer line and tap is cold. Turn the pressure down to a couple PSI, and vent the excess pressure from the keg. Chill the bottles down as cold as you can. Burp the stopper just enough to keep the beer flowing. Honestly, I've become much lazier, I mean pragmatic, lately, and just shove a piece of tubing into the tap and fill without the stopper. If everything's cold, and the pressure is low enough, it works just as well. Just fill until beer, not just foam, is coming out the top. Cap immediately.
Good luck in the competition!