Instead of an inner cover, get a piece of cheap flooring vinyl (like you use in kitchens) and cut it so there is a 3/4inch gap all around when you sit it on top of the frames in the top super (we call it a hive mat). Make sure the vinyl is the type with the thin felt-like backing, with the felt side down on the frame tops.
Bees case beetles. Beetles run and hide in nylon felt. Beetle legs have barbs. Beetles get caught in felt. When beetles caught, move mat to freezer for couple of days. Frozen beetles dead. Bees happy. Bee keeper happy. No messy oil. No chemicals.