What you are describing could be very dangerous. The purpose of the transfer switch is to prevent two AC power sources from powering the same circuits simultaneously. Unlike DC, AC power has to be in phase - the sine waves must be perfectly matched if you attempt to feed two separate sources into the same circuits at the same time. Since your inverter has no way to sync with shore power, that would never be the case when your inverter is on.
I believe what you are suggesting is to make sure the inverter is turned off anytime you are plugged into shore power, but what if you forget. There is also a scenario where your inverter could back-feed into the power grid and create a dangerous situation for someone trying to make electrical repairs somewhere else. Sure, your inverter has limited output, but that is still enough to create a dangerous situation.
Here is a link to a video I created a while back that shows the inverter prep circuit and ATS on my Reflection.