Thank you for your contribution!
I will test this out soon(ish, I hope...), and merge it if I have no further remarks!

I tried your code out but:

1. I couldn't see the toggle (well, a marker does appear named bm-popup-set but it's untoggleable)
2. But the distance does appear in web console + in that bm-popup-set marker)

The bm-popup-set marker is not my own, the 'Distance Measurer' addon alone also makes it appear. I only reworked the code for the moment.

If the toggle does not appear, it is possibly due to the fact that it is not positioned in the same place for everyone. When I coded the button, I gave it a position (modifiable in the code) but friends of mine pointed out to me that depending on the size of the screen, the browser or the zoom applied to the same browser, the button was not in the same place, or even behind the BlueMap buttons. Which can be very annoying since it is the one that allows you to enable and disable the distance measurement.