Does the inverter sync with main? If it doesnt beware you will have spikes during switching and this can kill electronic appliances!! Also note that there are no "fast" ats systems as such. You can get around 20ms in mechanical switching but those can cause lights to flicker.
A proper ATS have both electrical such as mechanical interlocking so that it cant engage both sides. Skipp all those junk chinese ones out there because many of them dont come with proper interlocking.
What you look for is a mechanical relay where you have 2 relayse with aux ports and mechanical interlocking
You basically look at the same as is called reverse contactor for motors