Page action
I believe that a page action would fit this extension better, as it works similarly to the existing Send Tab to Device, something like Send Tab to Container.
I've realised that someone already made this suggestion, by looking at the reviews on the addons page, but you rejected it based on Firefox guidelines:
Page actions are for actions that are only relevant to particular pages. If your icon should always be available, use a browser action instead.
I have a different interpretation (and I am hoping this will make you change your mind): page actions are specific to the page, and are not meant to affect anything but the displayed page. There is a difference between the action Bookmark this Page (action which is, by the way, always displayed, apart from the new-tab page) which refers to the active page, and the Bookmark Menu which relates to the browser itself. This way, the potential "Send Tab to Container" should not be a browser-wide setting; it is specific to the page, and its contents depend on the page itself: for example, you cannot send the page to the same container (A > A). Finally, if you were to follow the rules scrupulously, the Send Tab to Container action should not be always available: it should not be displayed on New Tab, about:* (preferences/addons/config...), or Private pages.