Popup time picker for custom card

Hey,
I am currently working on a lovelace card, which should set an input_time entity. So I would love, that when you tap a specific element on the card, a time picker pops up. Unfortunately the native <input type="time".../> is unsuitable, because of presentation issues. Is there any way to create a custom “more info” popup? browser_mod does not seem to be able to do more than showing lovelace cards in a popup, and would need to be configured outside of the lovelace card, which I would like to avoid.
Thanks!