Tested under Linux Lite 3.3 (based on Ubuntu 16.04).
Steam in a 32-bit bottle will only download around 160 MBs when Battleborn is installed, then complains that it is missing the executable when you try to launch Battleborn.
Crossover does not have a 32 bit version available, however, I installed WINE from the Ubuntu repo and attempted a 64 bit install. Battleborn installs in its entirety, but an error pops up when you try to launch it. I'm not at home at the moment to capture the error, but it is currently a show-stopper for me.