2 inverters to a 3 position switch?

You could, yes. But I'd agree that just using the bigger one would be much easier, safer and less complicated just to save a few Ah/day.

300-watt invertet for what kind of load? TV?

If you can power all your low value AC loads from one power center why not dedicate the small inverter? Let the inverter operate like a UPS when.plugged in to shore or gen. A continuous 300-watt load is not insignificant.

The treat the larger inverter install lile a normal install.

you can do it with ATS transfer switches , 30amp double double throw relays in a box, cost $70-$99 each, you will need two, plus plugs and wire for connecting them
and you will need a 2way rocker or toggle switch too control them, so only one inverter can be active ( it has to be either-or ) not two switches , it has too be only one can connect
this is only switching the outputs, so you need other switches for remote on off

safer and easier to mount them side by side and just move the shore plug
easier still to just use ONE inverter, get one with load sense, that turns on when used and has super low idle current needs when not being used

I think this is a receipt for disaster and really unnecessary additional expense.

How long will it BE before somebody turns on the wrong appliance when the wrong inverter is active and blows it up ??

I've solved that problem. I run the generator whenever AC is needed.
Easy.

Once you've bought a 2000w inverter, why mess around with a 300w inverter. Losses, will be negligible given the description of what you want to do.

To answer in a literal sense: Yes but only if wired properly. Since you are asking the question, get some help.