You have a bad connection somewhere. The 110 is getting loaded down by the heating element which draws a lot of current.
If it actually draws enough current to drop the 110 down that much you would pop a breaker. At least one would hope.
A few voltage measurements are in order and depending on how easy it is to get to you starting with either the transfer switch or the AC breaker panel will work.
You most likely have a load shedding circuit board inside your AC breaker panel which could be sick.
To help eliminate the transfer panel start the generator and see if the problem clears. If it does not the transfer switch could still be the culprit and in any case inspecting and tightening the connections is always a good plan.
There is a history of certain transfer switches and in some cases of fires. My transfer switch is easy to get to living in the electrical compartment of my rv. Next to the cord reel on the back.
If getting to that is not as easy just take the cover off the AC breaker panel and put a voltmeter on the breaker that feeds the water heater. If the voltage holds solid there take a look at any diagrams that may be on the back of the panel cover.
While looking inside your breaker panel you may/should see a a circuit board with wires along the edge.
The wires carry the voltage into relays and back out again of course. The relays can become defective and I have even seen that board catch fire. Fortunately in a metal box it died out quickly.
I find it interesting your panel is reporting some sort of message and would like to see what it says and looks light just to save that information.
As for the refer working on propane. It only needs 12 volts and propane to run. The 110 is for the heating element when switched to 110.
Are you on shoreline with 50 amps? Or 30 amps? Dogbone adaptors can fail and cause some strange symptoms too. But a voltmeter will tell the tale quickly.
Just knowing the paths that power takes can be difficult to chase for sure.
Certified Senior Electronic Technician, Telecommunications Engineer, Telecommunications repair Service Center Owner, Original owner HR 2008