Yes, this would be a great addition to battlegrounds. I believe it would make more people want to play as well, because of the competitiveness. Maybe add a /stats for battlegrounds? Hopefully it would not take up too much server space, and cause lag. But again, if possible that would be a great addition.
Yes, I think this could be an interesting feature to implement for Battlegrounds, of course we can focus on the Leaderboards, however under the current rankings there could be a small section saying what your kills are and what your rank is against all the other players currently in the server.