Make extra_badge on light object to run a script on click event

Configuration
1

Hi, I would like to add a badge on my light that I can run a script on click. At the moment my config is:

light.lamp:
  friendly_name: Lamp
  icon: mdi:ceiling-light
  extra_badge:
    entity_id: script.bedlightwhite

But in this way I have “off” label and on click appear a popup with “Activate” button. I would like to active my script already on “on click” event on the badge with a custom label instead of “off”.
How can I do it?

Thank you very much