There have been a number of Tripp-lite inverters with the same symptom. The problem seems to be with the "load sense" dial setting (you can adjust) or the "load sense" potentiometer itself (requires that the inverter be replaced).
The "load sense" dial can be adjusted using a very small flat blade screwdriver. Refer to page 8R in the Tripplite owners manual for adjustment and page 4R for it's location on the inverter (it is #9 in the reference). There is now a plastic guard built onto the inverter that makes it difficult to see the "load sense" dial, a small mirror may be helpful.
To adjust, have the inverter set to "Auto", 110 volt circuit breakers "on", and slowly turn the "load sense" dial clockwise until the microwave controller stays "on". If the "load sense" dial is already turned all the way clockwise, then slowly turn it counterclockwise until the microwave controller stays "on". The "load sense" dial is very sensitive, the reason the microwave controller cycles "off and on" is because the inverter doesn't detect the minor load that the microwave controller puts on the inverter. The inverter "searches" for a load to tell it to come "on", the microwave controller just doesn't draw enough enough power to keep the inverter "on". If the "load sense" dial is working properly, turning the "load sense" dial all the way clockwise will fix the problem. Sometimes, turning the "load sense" dial slightly counterclockwise will compensate for what I think is a poorly built potentiometer.
Another quick way to see if it is the "load sense" dial is to plug a 110 volt appliance into an outlet, turn it on, and then try the microwave. Provided the appliance you plug in doesn't draw too much power, the microwave should stay "on".
Hopefully this helps. Were you informed at delivery how to "reset" your inverter if it shuts down because your battery voltage is low?
Here is a link to the manual; http://www.tripplite.com/support/manual ... delID=2569