I was thinking at first that it might be a bit of a pain switching back and forth from one side of the screen to the other with the mouse cursor
I think you're misinterpreting the use I'm suggesting. So you can left click and hold to "grab" one side. You then slide up or down to tilt. The side that tilts is based on the side your mouse is on (so if you don't have an even number resolution, you'll need one). Since it can go up OR down, you'd be able to change everything from one side if you so desire. Just if you move to the other, it would behave like one would expect. You could even make it based on how far to one side (so closer to centre makes it turn way faster than if you're on the edge of the playing field).