And the state of it persist across page loads or tabs?

No. It might, depending on what your browser does, but it's not in the web standard.

But you can have a button that saves your state when you enable javascript, and doesn't save your state (but still works) when you disable javascript.

edit: I think it is possible to save your state on the second click. So the UX is: you have 3 options with a slide. You click one of them, the page theme changes, and the option icon becomes a padlock. You click on it again, and the option is saved.

It seems to be a limitation that without javascript a single click can't change a switch and do something else--make a request to set a cookie. But you can do changing style on first click, then setting a cookie on the second. Here's a demo (written by Claude) (it doesn't work without server, just the HTML part) https://jsfiddle.net/r134vgo7/3/