Could you please add the Lobby ID to the Game Lobby screen. Is this something someone could build a mod for?
Here you go. Unzip in your mod dir and enable with PAMM. Only casually tested, so be warned. Also it isn't obvious, but you can double click the shown lobby ID to select it and copy-paste it wherever.
Laziness mainly. This mod only took about ten minutes to make since I already knew where all the pieces of the puzzle were located. Getting it PAMM-ready requires I take care of edge cases such as proper behaviour with the local server, making a dedicated thread in the mod forum, creating an icon, and probably stuff I am forgetting. And with "require", I mean require by me, as I like my mods on PAMM to be polished. Since no other people replied with an "I want this too" to this thread, I figured the request was unique to you, so I decided to just post it here as is. Feel free to do whatever you want with it though.