I've been using and abusing inverters for decades on different vehicles and assuming you have 100 Ah battery, you might power 1000 w microwave out of it for couple of minutes.
Too many variables to be specific, but first thing to check is to make sure inverter has good cooling flow. With power close to max output, it could overheat faster than battery will drop the voltage to cut it off. Length of cables will play role as long cable will create higher voltage drop.
Over the years I managed to get 800W coffee maker and 800W toaster.
Even you could calculate that running 1200 coffee maker for 6 minutes and 800W 9 minutes takes the same energy -inverters tend to work much better at lower power draw.
Beware that at high load - the voltage will drop below 110V, so heating time will need to be extended.
My favorite experience was using 1200W inverter to run 1500W oven on my boat with 30 amp alternator and 60 Ah battery.
When I used oven only with engine on, it would routinely cut off after 7 minutes, what was due to overheating.
Boat had good cooling and San Francisco Bay is never really warm, yet overheating still was main problem.
The 30 amp charging breaker never flip.