What about the simple solution? When you log in you are placed on the side of imbalance? Perhaps a message telling you what country you're on. Still give you the option to change sides once a day, not once a log on. Then you would see who truely has squad/country loyalties as opposed to those who do not care. Perk system can stay as is.
How many online play and do not frequent the BBS to complain about what country they are on? It's a question that is tough to measure. Only the guys who are really into this game are on the board, any idea of what percentage of player base that is?
Player community has this country loyal mentality but this could be because of the way the game is setup? You're always logged into the same country. What would happen if you logged in at the same time as three of your squaddies and they are all Bish instead of Rook today? Are you going to switch sides? You know it's not a squad decision they are Bish. So does it matter?
