It's a workaround, but you could enable a virtual desktop in winecfg. The
game won't be completely fullscreen, but it won't mess with your screen
settings either.
Turning the second screen back on through a tool like lrandr after having
started the game works too.
This is more of a video driver related problem than a Wine one, though. If your
driver comes with a settings tool (think the ATI Catalyst Control Center for
example), open it and see if you can find a way to always keep the second screen
enabled.