It just depends on what the goal is.
If you pool all the kids, then yes, there is a greater chance that larger branches might have more winners; but every kid in the city has the same chance, no matter which branch they go to.
I know attendance dropped when there was only one prize, but that was based on a fallacious perception--every individual kid absolutely had the same exact chance to win that prize. It's just the branch that had lower odds. And the kids from the smaller branches MAY have thought, "my neighborhood won't win, so that means I won't."
BUT they also may have thought, "there's only one prize for the entire city--everybody's odds are low, and since mine are low, it's not worth my time."
Perhaps the incentive of winning the prize is the only thing that made the kids from those neighborhoods read a book, and the kids from the neighborhoods with high participation would have read those 5 books anyway.
I was going to say that since the goal is to get kids to read, you should go back to the prize structure that worked, and not worry whether the kids at the larger libraries had a lesser chance to win, since you're still getting them into the contest.
But, since there was only one prize, you don't know whether that would have worked.
The cool thing would be to pick the smallest library (let's say Library A=500 kids; Library B=1,000; Library C=2,300) and come up with a Decent Prize that you can do multiples of, and then you have one prize for every 500 kids in each library, rounding off (Library a = 1 prize; Library B=2 prizes; Library C=5 prizes).
Then every kid has a roughly even chance, but the kids in the smaller neighborhoods are less likely to think, "Oh, we're outnumbered by all those wealthier suburban neighborhoods, so I'll never win."
EDITED TO ADD: In order to achieve your goal--to make kids feel like they'll get something good if they read--I think you do need to have a plan that means there IS a winner at every branch. Kids at Library A aren't going to feel all that invested if the winners list doesn't include someone from their branch. But the same could be for Library C.
in the pursuit of "fair," you shouldn't lose sight of the idea that your aim is to get kids to read.