I think it might be an issue with the built in ground fault detection in the unit. A fault on the grounded leg would probably screw with the electronics.
I'm assuming these are EVSEs. They should provide downstream GFPE-type protection (CCID20, say). But that shouldn't care about the L-G voltage or if one of them is 0V.
They may (should?) also have an EGC assurance test, in which they intentionally try to pass current L-G (but not too much, as they aren't supposed to trip upstream GFCI). So if the L they use for that is actually N, there would be no voltage difference to drive a current, and the test would presumably fail. That problem can be avoided just by figuring out whether they use L1 or L2 and landing the N supply conductor accordingly.
There's also the issue that the EGC assurance test may be designed around 120V L-G. So something might go wrong if provided with 240V L-G. E.g. a component gets more voltage or more current that it is designed for. That would be a deal breaker. Of course, the circuit could easily have been designed to handle both 120V and 240V L-G, but if it has only been designed for 120V L-G, then that's moot.
A final possibility is that the EGC assurance test will work fine at 240V L-G, but that it will pass twice the current intended, potentially tripping upstream GFCI. In which case it would only work in an installation design where upstream GFCI is not required.
So apparently I was also at least 48% overconfident in my first response. : - )
Cheers, Wayne