My guess is that it's high inrush current to the microwave that's causing you problems.
Microwaves, and most other modern appliances and electronic devices, often have very high inrush current demands. These can be 5, 10, 50, or even 100 times the average rated current, though only for a few milliseconds.
Breakers are usually designed to take inrush current as much as maybe 6 to 8 times their rating before tripping.
It could be also that your breaker(s) has/have deteriorated, as they do over time, i.e., they are unable to carry rated loads.
About the only thing you can do, IMHO, is to only start up the microwave when EVERYTHING else is turned off.