As you found out, the breakers or fuses, do not protect the inverter just the wires. The inverter has internal fuses, perhaps as many as 16 or more. But even they can not always protect the inverter. I had 2 inverters fail on sudden overloads, they did not shut down automatically, nor did they blow any fuses.
They just let out the magic smoke.
I currently have a 3000 watt - 6000 watt surge MSW and it has worked fine, except for once when I tried to run my cruisair AC, that fried it instantly, burnt out all 16 DC side mosfets, mosfet gate resistors, 4 mosfet controllers. Took about $50 in parts and had to rebuild burnt copper traces. The DC mosfets switch off and on at a very high frequency pumping up the low DC volts with transformers. Then it is converted to AC power. If the mosfets get stuck on, then they get all out of cycle with each other and the whole low volt DC side shorts and burns. A single shorted stuck on mosfet will destroy all the other mosfets. It is like some of these manufacturers use the cheapest parts they can find and build them to run on the ragged edge.
I noticed recently my rebuild actually works better than OEM, because I ran a pump without a start capacitor, and it shut down safely the inverter. Then I added a start capacitor in the pool motor pump and it worked ok. I did upgrade the low side caps to 25 vdc from the 16 vdc ones, and also choose good quality mosfets with bigger gate resistors. I never plan to run my AC off it ever again.
I wired my inverter into the boat's AC system, and using relays and manual switches, can select from shore, gen, inverter power. When the AC comes on, it breaks open the circuit to the AC automatically so it can never even try to start. If the gen begins a start cycle, it breaks off the inverter relay, so no power gets intermingled, Same with shore, if shore power is present it breaks off inverter and generator. So the priority is shore, then gen, then inverter. All done with various types of relays. Some relays are 4 pole double throw. With twin 30 amp power, you have to account for 4 power wires, 2 hots and 2 neutrals to be switched simultaneously.