Thank you! It was actually Chrome releasing the ability to open a sidebar using a context menu action that sparked the idea.
I absolutely plan to support firefox. I figured I should get things working well in one browser first before looking into porting, but consider it on the near term todo list.
Sounds good. One problem that you will likely encounter in FF is that you can't open the native extension popup (the one that is attached to the action button) programmatically. That severely limits how you can present your UI, with the most obvious choice being injecting an iframe with the extension UI directly into the body of the page. The other choice is opening a new popup window, which is less convenient than seeing things side by side.
Just FYI: manifest v3 extensions are already cross-browser. They are all based on the WebExtensions API. However, browsers have very specific constraints that require complex alternative solutions (e.g. Safari does not allow reading httpOnly cookies, so auth can't be handled easily).
1. using the new Chrome Sidebar API is a great choice!
2. selecting actions to show in the context menu is also a convenient customization idea
Are you considering porting it to FF? Other than the Sidebar API it should be a very straightforward process.