Maybe there is a better way, but I'd just use a temporary state. Have the button update the temporary state when clicked, and based on that state set the background color.
Here is a demo app:
dark-mode-test.json (6.1 KB)
I imagine it would become a pain to update all of the various components colors. I'm currently on the "Team" plan, so I don't have access to the theme stuff that comes with the "Business" plan. Might be easier with that 