Should this have been a neutral problem it would have shown up long before the pump test.
It was without a doubt the inrush of current due to the starting of the motor. To start a 1/2 horse power 120 volt motor the motor will have an inrush of about 58 amps which is far more than the small generator will deliver.
When the motor started the load on the generator caused the gasoline motor to load and speed up which in turn caused a voltage drop followed by a surge in voltage that the transformer could not handle.
Once again, these small generators were not designed to be connected to a house load. The proper way to use one is with the use of cords plugged directly into the devices (receptacles) on the generator.
On a side note, I have two small generators. I have horses that need water during an extended power outage (24 hours or more). I use the 7500 watt generator to power the well pump for water at the barn. The other generator is used for emergency lighting at the house. This lighting is a table lamp plugged into a drop cord ran to the generator outside. I have no transfer switches connected to my house at all.
A power outage is a very rare event and one in which my everyday way of living is interrupted and one which I live on a limited amount of electrical use. I do not try to watch TV or surf the internet during these times.
Should I for some reason or another need more than a cord during a power outage I will purchase a generator big enough to handle the load. What I wouldn’t do is try to haul 17 ton of rocks on a half ton pickup truck or use a portable generator to power my house. Think of it this way and see if it doesn’t make a little more sense than just saying I have a generator and I want to power up this and that during a power outage.