240 volt not supplying 120 over one leg and the neutral
Time for a nightlight test

Hit a dollar store and get two 2-prong extension cords and two incandescent nightlights (incandescent, without light sensor, straight switch-only). 

On the extension cords, lop off the plugs (pointy end) and throw the plugs away, we are not making a suicide cord today*. 

On the cut ends, split the two wires for about 1 foot (300mm).   Then strip about 1/2" (12mm) of insulation off each wire end.  The ribbed side of the cord is neutral, the smooth side is hot, if possible mark them white and black respectively. 

Test the nightlights, then plug them into an extension cord socket.   Now you have a test cable.  

With the main breaker off, land both extension-cord neutrals on the GFCI-breaker neutral (or the neutral bar if testing with a plain breaker).    Then land one extension cord "hot" on hot L1, and the other cord "hot" on pole L2. 



Power up and see what happens to your voltages

Make sure both nightlights are on (or should be on).  I have a feeling one might not light up, that's why you test them first.

Now go back and do your voltage tests again.   I bet whichever 120V supply was "slightly lesser" in voltage now goes completely away. 

That would be phantom voltage, which is picked up by capacitive coupling from adjacent 'hot' wires to a 'disconnected' wire.  There's no appreciable current behind it, and it can't do any useful work**. It is almost "static electricity" and is very easily neutralized by the slightest load (our nightlights).***  

So now, I think the symptoms will be much clearer and easier to troubleshoot.  

The trouble may be in the main panel. 



* If you have an actual use for a cord-with-plug, then buy an extension cord long enough for both that and this, and cut at the appropriate point. 

** unless tuning AM radio frequencies into a crystal earpiece is of interest to you. 

*** I prefer nightlights to 22k ohm resistors because Radio Shacks are getting hard to find; plus nightlight cords have dozens of other uses in electrical troubleshooting.