Even on Lichess web site there isn't a LIVE ranking of all BOTS: bullet, blitz, rapid, classic and correspondence.
To know how ProteusSF is scoring after each game, since there isn't the ELO after the name on the clock frame, I've this workauround: I open Lichess ProteusSF page in background and auto-refresh it every 60 seconds, but it would be much better to have the overall ranking in BSG in a fixed frame as you have done for Tournament Results. And all Lichess users would choose BSG as client for Win, Mac and Linux
...Something like this table taken from Playchess