Electronic – Can a solar inverter be damaged if installed capacity is much larger than demand

Electrons are fungible — you can't tell the difference between one electron and another. Therefore, no actual switching is required — your power meter simply measures the net power consumption of your house, which is the difference between what your loads require and what the solar array is providing.

In some jurisdictions, if that net value becomes negative — in other words, you're generating more power than you need — the meter does measure the power that you're feeding into the grid separately, because the rate at which you sell power to the grid is not the same rate at which you purchase it.

I don't have experience with the specific models you are referencing but I do have experience with phase locking AC power sources. The bicycle analogy above is exactly how mechanical generators work and with solid state power inverters to the point of the inverter "pushing" on the grid however you don't want the grid to ever "push" on your inverter. The inverter watches the 60hz sine wave from the grid connection to trigger the driver to keep the output in phase with grid power. This is going to be a high impedance detection circuit so any source should do the job such as your pure wave inverter off a car battery. The listed tolerance on the sine wave is well outside the normal tolerance for mains. How precise is the frequency of the AC electricity network?

While connected to the grid it is critical the inverter is synced to the grid frequency and phase as tight as possible to prevent the inverter from seeing the grid as a load. If the grid were to have just gone down and your system started automatically the phase could be 90deg out of sync when the grid power is restored which would be worse than shorting the output of your inverter. This is why they inverter relies on an external clock source and shuts down if one is not present. Disconnected from the grid you have a self contained power system and the phase and frequency relationship to the grid is meaningless but the frequency does need to be close to 60hz for any devices you maybe running on that power.