Personally I think a bio contest wouldn't be a good idea, it in no way tests the persons RPing skill and a lot of people can get others (non-Rpers) to do it for them which doesn't seem right. Not only that, it will encourage people to submit/drop shit loads of bios just to try and win and it will create more work for the Mods checking them.
A battle test of the month or something like that seems like it could work, imo ofc. It will not only encourage the proctor and testee to put more effort into their replies, it will push people to take up more challenging tasks instead of just fighting in the battle arena, bringing with it more activity and an increase in RP quality due to people wanting to win.
Also I don't think any of these contests should be decided by public voting, (if Trump being elected has taught us anything, its that people don't know what the **** they're doing) because it just becomes a popularity contest and we know how that ends.