I feel that an error alert is not the best use of the menu-bar popover. Most applications I know of use this mechanism as a menu: The user keeps the icon in the menu bar because it shows some sort of information, and clicks on it to access further information and actions.
I think the typical OS X model would work something like this:
Menu Bar Icon: has multiple states, to reflect current status. You could have an icon for "Good connection" and another for "connection issues"
Popover: The popover, as in the mockup you provided, should activate only when the user clicks the menu bar icon, and should be focused on an action. If you offer a way to resolve the connection, include it here. If that's all the popover includes, make it very small. Otherwise, include other info/actions that are useful to your user.
OS X Notification: If a bad connection status is a time-sensitive or very important thing for the user to pay attention to, AND the menu bar icon is not prominent enough a way to communicate the status, use a notification. The notification should also include an action the user can use to resolve the issue.