That is why I mentioned it earlier, the voltages OP mentioned add up to too much line to line volts which is pretty much never an customer problem if you have no customer transformation going on.
Long marginal sized service conductor or feeder can also mislead you to thinking about bad neutral, particularly if load severely unbalanced and it wouldn't take much to get the five volts of imbalance that the OP is seeing, but on split single phase source you still can't add line to neutral voltages at the load and end up with more than the line to line input from the source (at least not without a boost coil somewhere in the circuit).
I have only saw this one other time where the voltages didn’t add correctly as you stated.
We had a single phase transformer that served a house doing about this same thing. Electricians came by multiple times. Yes, our guys said bad neutral in the house also. The guys resqueezed everything outside.
Finally had the guys pull the service leads out of the XF and check the XF by itself with nothing on it.
Same voltage readings. 124 on one side, 130 on the other side. 254 across..
(I just used those numbers, I can’t remember exactly what the voltages were)
Took the XF down and changed it out. All was well.
We TTR the XF and one coil was off just enough to give higher voltage. Junked that unit, (think it was a 15kVA) even though it was within the 10% tolerance.