What's the trick to beating level 4-9 of Escape Goat 2?

So I just can't figure out what to do in this level. Here is what I start off with:

The two buttons in red cause the walls to move inward, resulting in the wheels starting to roll. The buttons in green toggle each other, and cause the circled platform to move.

Hitting the left button doesn't seem to do much for me, since the wheel that moves as a result just falls out the bottom of the map. Throwing my mouse up to the ceiling doesn't do anything either.

But, by hitting the right button, and then timing it right, I can manage to throw my mouse up onto the right side of the ceiling, allowing it to hit the green button. Unfortunately the platform doesn't have enough time to get low enough for me to jump on it before the wheel hits the second button and the platform starts moving upward again. When my mouse comes back around and hits the button for a second time, there's no way to reverse the platform to head back up, since I can't access the second button.

I seem to be missing something obvious here. What is it?