Shelly, my idea was that you'd put your BHSE on the top shelf and acquire the cloth cover of your choice. I'd just use a large enough cotton towel but you could choose whatever suits you. When the BHSE is on, the cover is off. When you are done listening to the BHSE, let the tubes cool a bit. I'd think that 5 minutes would be enough. Then dust the BHSE and cover with the cloth until the next listening session.
This solution is simple, effective and easy to execute.