Bulbs, especially cheaply-made LEDs can fail internally in various ways. See Understanding and Preventing LED Failure.
However, this particular circuit issue of yours sounds complex, and may be the result o. I would do these things, in this order:
(1) Try completely different, new, premium, LED bulbs from a different manufacturer. (Are the ones you are using cheap, imported, and have you tested each one individually on a separate circuit?)
(2) Test each outlet/receptacle in the circuit using a common electrical outlet tester. You may very well have an outlet that confuses neutral and ground or some other strange combination that is messing this thing up.
Other questions that may need to be asked in order to solve this:
(BTW, what type of instrument are you using currently to test the voltage at the microwave outlet?)
(a) Are there other issues in the wiring of the circuit itself, i.e. things wired in series, or so forth?
(b) Can you use another circuit for the microwave?
(I don't have enough points to make comments yet---so this is entered as an answer!)