LED lights cause appliance failure
No, because you have worse problems than that.

The situation of "LEDs make it fail/act weird, but even one incandescent fixes it" indicates only one thing: a powered switch of some kind, a dimmer, lighted switch, smart switch, motion sensor, something/anything like that.  

Obviously, nothing like that should have any relationship whatsoever with your microwave, but the evidence is clear that it does. 

And the problem is not as simple as identifying the smart device and changing it to a dumb switch.  Something is fundamentally, seriously wrong and dangerous with that circuit, this is just the "canary in the coal mine".  

Also beware of multi-wire branch circuits -not that there's anything wrong with them if done right, but they are much more sensitive to being done wrong.

You might also change the breaker  to GFCI - it will help motivate you -- I mean will help detect faulty wiring.