We do get a lot of non-regulars at the contests I've hosted. One rule of thumb is that I kept the "qualification" round limited to two hours. I ran from 9pm to 1am (or later), but you could only qualify for the contest between 10 and midnight. This left room for other singers to sing just for fun, and allowed for contestants to do a practice song if they wished.
For the qualifying weeks, the judges were myself, the bar owner, and the bartender. This was just to decide which two singers would advance to the semi-finals; at the end of the night the three of us would sit down and pick two. We did our best not to favor regulars or friends...
Down to the final rounds, the bar owner brought in some people he knows from Chicago, in various music-related fields, gave them free drinks and probably some cash...
There are *always* arguments from the vocal minority about how it must be rigged etc, but most people just enjoy watching the contest and seeing who leaves with that grand prize.
The last one I did was the only one where we did a $1000 prize, and the girl who won is only an occasional customer. She has 3 small kids and really needed the money - and based on her singing she really deserved it as well. People appreciated that, and didn't argue much about who won. It turns out she has tried out for American Idol when they were in Chicago - she didn't get in, but she was good enough to at least get an audition...
Like I said before, you might ask your regular singers what they think about a contest, without mentioning any specific prize amount, just to get a feel for how it might go over.
As for the duration, we went with 12 weeks (10 qualifying then semi/finals) because shorter contests didn't build up the momentum and excitement about the finals as much. You can give each qualifier a business card with the Semi and Finale dates written on the back to ensure that they, and their "cheering section" who will of course be buying drinks, will remember to be there. This also gets a business card for yourself out there that they are sure to hang on to
If you print your own at home like I do, you could even do special cards just for qualifiers, that mention the bar, contest date/time, etc... I didn't do that but I only thought of it just now

That would create a memento they will probably keep, and if they ever need a DJ...