EDIT: This has been fixed and already available on Chrome version of this extension. I'll verify the changes for Firefox as well and publish it here soon.

Glad that you like it :-)

When the extension was created, the reason for not applying the CSS automatically was security concerns. The reason is explained at https://github.com/webextensions/live-css-editor/issues/2

It appears that what you are asking would be possible to do (with dynamic permissions API). I have added that as a TODO task at https://github.com/webextensions/live-css-editor/blob/master/TODO.md#improvements

Though it might take some time before I look into it.