Why is 700 Watt microwave shutting down 2,000 Watt inverter?

The inverter is shutting down because of low battery voltage, the same reason the truck low voltage alarm is going off. I'd test the batteries first, but you may also have a failing alternator.

And keep in mind the microwave puts out 700 watts, but it uses more than that.

 

What make and model of inverter, how is it hooked up and what size cables?

these questions all matter on what the real problem is.

 

Have you checked your battery connections?

 

One of two problems, either a worn out inverter or worn out batteries. Could be a bad connection to the batteries.

I don't think Internationals beep from low voltage. I drove one for years. I think that's the inverter beeping to warn you it's about to have to shut off. If the display isn't showing low voltage then its not the truck beeping.

 

How long are you running your microwave? None of them, even with an inverter, do well if you're running the microwave for more than 2-3 minutes. Are you cooking frozen dinners or something?

 

I'm assuming you are running the motor whenever using high-draw appliances like this?

 

Did you read the OP?

 

He never does.

 

Is this a new thing, or has this been going on from the start of use?

A 700 watt inverter can draw up to 3 times the operating watts on startup.
After start up, it will likely draw 850 to 900 watts operating rate.

If new, would look into all wire connections, clean any corrosion, make sure all connections are good and tight.
Inspect and test all your batteries. Make sure they are in good condition, this should include a static load test, and recovery time. Possibly you have weak batteries and can’t buffer the load demand. Finally test the charging system. If you can’t remember or don’t know when the drive belts have been replaced, probably a good idea to have that done.