In theory you can but the way you plan to embed it will determine its possibility or not.
Practically its going to be hard to do it in plain bubble without some javascript for the frontend UI widget.
If it was a simple inline embed widget i could see how you could do that with plain bubble in an iframe.
You could create the backend and host the messages with bubble.
The challenge will be the frontend UI widget.
You can’t open a popup from an iframe.
So its going to be hard and maybe almost impossible to have that icon floating on the page without some extra javascript file the user has to load.
@keith built a saas calendar widget in bubble. You can read more about his process and maybe learn something from that.