Have you considered braces of any sort?
Something like this
during the day, so you can still do stuff.
And more heavy duty and restrictive, like this
, for at night when you are sleeping.
Also, there is a technique I used, when I didn't have a vertical mouse, that lessened the amount of time I spent with my hand in a palm down position.
Make a fist or cup your hand and rest it on top of your mouse, like you were holding onto an invisible joystick, and move your mouse around that way. Only grab the mouse like you normally would, when you are ready to click. That will keep your hand in a more natural "handshake" position for a lot of your mouse usage.
There may also be some autoclick utilities, that will click the mouse if you stay in one place for a few seconds, eliminating the need to move your hand out of the "joystick" pose, unless you need to right click. I can't recommend one for you to try, though. Your mouse software might even have the feature already, so check there, first.