I've only done garden-bed shroom farming one time, and it was mostly a spectacular failure, but if I were going to try it in hugelkultur....
I'd prep a bunch of take the same species of tree that will be buried, and chip a bunch of it. Sterilize and inoculate it (Oysters are usually considered relatively easy), and allow the mycelium to run through it until you have a large quantity (garbage bags) of mycelium-colonized material. Then (timing will be tough here), build your bed and sprinkle the colonized chips throughout it, making sure to get a large quantity of them up against the wood. You might also want to consider adding a few trace ingredients to the hugelkultur bed to enhance fungus health--I believe powdered oyster shell was one of the ingredients?
Seems like the mycelium would stand a relatively good (still probably pretty bad!) chance of spreading throughout the logs.
+1 for fungi perfecti, really cool company. Their catalog is great for toilet reading, if nothing else