I have never done something like this before, but searching how to add a card inside a picture elements I found this Adding additional cards to Picture elements card maybe it can help.
As an alternative, using Browser Mod (GitHub - thomasloven/hass-browser_mod: 🔹 A Home Assistant integration to turn your browser into a controllable entity and media player), you should be able to use an icon that when tapped calls a service which opens a popup with the todo list card inside.