You should verify that between L M and H and N, there is 120 v for each position. I believe this will be the case. Then you can just connect L to M and H and have a single wire that provides power.
You should check the contact ratings on the FAN switch and/or the current for the internal fan, but what I would do is the following: Put a relay with a 120 V coil in there so all you have is a low voltage contact closure out.
Arrange this circuit to energize the relay when any fan position is selected and provide a potential free low voltage contact to the booster fan. This allows you to not be concerned with restrictions of running 120 V through weird spaces. Provide a connector on the back of the microwave.
Now you can run the 24 VAC control signal to the attic booster similarly as what is done in a home furnace.
In the attic mount a energy limiting control transformer typically used for furnaces in a large J-box with a relay with a 24 VAC coil. If you can't get an energy limiting model, put a 2 A automotive fuse in the secondary. Have that relay turn on the booster. Energy limiting transformers will survive a direct short across it's secondary forever.
If the fan center does not use an energy limiting transformer, get an automotive blade type fuse pigtail and add it in series with the secondary.
You can get all of this functionality (although a little overkill) in what's called a fan center. e,g,
https://www.google.com/products/cat...=X&ei=0oDNT-2SINGd6AHnmPDnAg&ved=0CK0BEPMCMAM
It would be nice, but not essential to power this J-box from the same circuit as the microwave.