So in the DJI GO 4 app I can assign different actions to this little joystick button. It says I can press it up, down, left, right or inward. However, it only lets me assign the first 4 actions. Is there a way to assign an action to inward press? I'd like to make it so that when I press inward, the gimbal switches forward/down.

No option for inward press. By default, that brings up the Intelligent Flight options. I use C1 button for toggling Gimbal forward/down.
Furthermore, FWIW: I use Left to toggle Portrait/Landscape, Right to select AELock, Up/Down to Zoom In/Zoom Out.

Unfortunately there’s only four options.

Hm, thanks for the idea to use C1/C2, somehow I didn't really think about those buttons
In that case it's not that bad that inward press is not customizeable.

Yes, you can customize the 4 options for the 5D button. Like eddieNoob mentioned on 2#, inward is for bringing the intelligent flight modes, and you can set the C1 or C2 for camera forward and down. Thanks.