You did download the new launcher succesfully?
Is the Launcher than not starting ?
or can you go through the entire Launcher process ?
(agree to terms, install directory, login then options screen with launch button)
and it isnt after that starting the Game after hitting Launch ?
if you do have a JAVA JDK installed .. please try to change the custom java path under options to that as a test.
could you be more precise here please ?

Yes, apparently. I added the beta code, selected the beta option. It updated in Steam.

Now when I click "Play", I get an option to Play or Play with Beta Launcher. If I choose Beta, I get the Steam error above. Nothing else launches.

If I go to the Steam/common/StarMade folder, and click "starmade-launcher" it comes up, with the terms, offers to install! (pointing at my Library/Applications folder, not the Steam already-installed version)... etc. I presume it works, I haven't tested that yet.

It looks like the Steam shortcut (or whatever you call it) is pointing at the wrong path. Please note I have the game installed on an external drive.

OK .. nothing we here can do about that i do assume.
we are only testing the Launcher itself and the game here ...
i will redirect this to someone who may have a clue about that steam client connection thing.

you are able to otherwise run and play the game with the new launcher just not able to start it from within steam ... just to make sure i did understand that right ;)