Unable to click on the menubar
When you set "LSUIElement" to true, it makes the application in question a semi GUI-less application.  While calling up the application in this state, it still shows up windows, but it does neither show its Dock icon nor its menu bar.

As to alternatives: The only real alternative is to have the application support this on its own (e.g. inside its own GUI).

Keep in mind that enabling "LSUIElement" (a.k.a. setting it to true) is not telling the application to hide its Dock icon, but rather tells the Dock not to show the application's icon while it is running.  Also GUI-wise there are two main processes running for the user: Finder and the Dock.  The first is merely an application on its own and will allow you to browse and alter the files on your computer and the latter besides showing and controlling the Dock icons controls the menu bar itself (not the menu items of each application, mind you) as well as the Dashboard and Launchpad among others. You can see its effect when running Terminal and type in "killall Dock".  It will effectively restart Dock and its depending items.