Somehow, I’ve only just realized that a badge icon can be used to display another entity status on a template card, even though it’s written in the documentation. I now use it to display the status of presence or motion sensors in the respective rooms.
badge_icon: >-
{{ "mdi:motion-sensor" if states("binary_sensor.0xb4e3f9fffeb689d4_occupancy") == "on" }}
badge_color: >-
{{ "green" if states("binary_sensor.0xb4e3f9fffeb689d4_occupancy") == "on" }}
