I would like to share my experience with FB login by Bluestack, I dont know exactly the step but try to go around as my method
1- Log in facebook by chrome in bluestack first
2- Uninstall the current version and install the 5.2.2, connect to FB account, then it will back to the loading screen and ask you to upgrade, close the app and open again, it still ask to upgrade, OK ignore it
3- Install the 5.5.0 version (do not log in, as I remember I do so)
4- Disable the Chrome (not uninstall), then open the game, of course the game cannot start, it will crash in my case
5-I close the Bluestack then open again, then enable the Chrome, and then open the game
6- Magically I can open the game with my FB account
>>> I already do this for my 2 accounts, so at least it work for my Bluestack, I use Bluestack 5 just FYI
Please try and hope It work for you