Do not use any parts of the oak as a mulch for otther trees because you can spread it if you do.
Id take it out especially if the oak its dead and extract the roots also.
The wood its good to use for fire or for any otther purpose except woodchips for mulch.
Aparently its impossible to kill armillaria and there is no cure for it.
Soil fumigations might work the best and planting resistant species of trees to isolate the infected spot.
Also,if you like to eat Armillaria mushrooms ,like i do,be carefull where you throw the parts resulted from cleaning and even the water in wich you wash them.Do not throw it in the garden or on the compost pile .