MacOS – How to show the macOS menu bar on fullscreen mode

I don't believe this is possible in Lion.

For your specific problem, I'd just use the keyboard shortcuts to move to a new tab; it's much faster anyway.

To move to the next tab, command-shift-}. To move to the previous tab, command-shift-{.

• Save the following AppleScript to a file named fullscreen.scpt:

  use framework "AppKit"
  use scripting additions
  
  repeat with runningApp in current application's NSWorkspace's sharedWorkspace's runningApplications()
      if runningApp's isActive()
          set frontApp to (localizedName of runningApp) as text
          exit repeat
      end if
  end repeat
  
  tell application "System Events"
      tell process frontApp to set isFullScreen to value of attribute "AXFullScreen" of first window
      if frontApp = "Finder"
          tell process frontApp to set value of attribute "AXFullScreen" of first window to not isFullScreen
      else if isFullScreen
          do shell script "lsappinfo setinfo -app " & quoted form of frontApp & " ApplicationType=Foreground"
          tell process frontApp to set value of attribute "AXFullScreen" of first window to false
  
          (*fix to make sure the menu bar is not stuck*)
          delay 0.42
          tell application "Finder" to activate
          tell process frontApp to set frontmost to true
      else
          do shell script "lsappinfo setinfo -app " & quoted form of frontApp & " ApplicationType=UIElement"
          tell process frontApp to set value of attribute "AXFullScreen" of first window to true
      end if
  end tell
  

• From terminal, compile it to an application with the following command:

  osacompile -o "/Applications/Full Screen.app" fullscreen.scpt
  

• Open the Full Screen.app's Info.plist (e.g. vim '/Applications/Full Screen.app/Contents/Info.plist') and add the following to the dict:

      <key>NSUIElement</key>
      <true/>
  

• Add Full Screen.app as an exception in System Preferences > Security & Privacy > Privacy > Accessibility.

• Launch Automator and create a new Service.

• Change "Service receives" to "no input in any application".
• Add a Library > Utilities > Launch Application action.
• Configure the action to launch the previously created Full Screen application.
• Save the service as Full Screen and close Automator.
• On System Preferences > Keyboard > Shortcuts > Services, scroll down to the bottom of the list and the just created Full Screen service should be listed there. Associate an unique Command shortcut for it, like Shift+Command+\ or Command+F11 for example.

This creates a shortcut to cause an application to enter full screen while removing the menu bar, or to exit full screen bringing the menu bar back. It provides an alternative full screen shortcut!

For application-specific full screen launchers, check my other answer.

Caveats

There may be some disadvantages and/or misbehavior using this approach:

• It works by setting ApplicationType=UIElement, which causes the application icon not be added/highlighted in the Dock and make the application inaccessible via Command+Tab. The Command+Tab issue was reported in comments, I didn't notice it since I mostly use the Mission Control overview to change between full screen applications.
• It may not behave as expected for some specific applications, I've noticed issues with the Activity Monitor application (which is generally not used full screen anyway) and there's a report on Chrome, which I didn't try since I use Firefox and it works great.