So, it's a bit finicky, but here's how I managed to get it running in a half-decent way...

a) You'll need winetricks. Install it. I leave the hows up to you, as there are many options available.

b) Run the following commands before starting the game:

winetricks windowmanagerdecorated=n
winetricks windowmanagermanaged=n

These will fix the game's apparent inability to scale its menus properly, among other things.

c) Run the game. Start a mission. You'll now probably see your desktop instead of the game's UI, but don't fret. Alt + Tab to any other window other than your game window.

d) You should now see the scaled-down game window on your desktop. Click on it and it will scale back to fullscreen.

e) Play the game to your heart's content.

f) Rinse and repeat from c) to e) every time you start a new mission (unfortunately).

g) Run the following upon exiting the game, if you're using a shared prefix and don't want to potentially mess up stuff with other games:

winetricks windowmanagerdecorated=y
winetricks windowmanagermanaged=y

You could also do a bash launch script which contains b) and g) ...and c), or at least launching the game with Wine.

If anyone finds a better way to get around these things, do tell - I'm not completely happy with this approach either.