I was originally going to make my HEX moveable between the HLT and BK so that I can use it as an immersion chiller, similar to this guys setup:
https://www.homebrewtalk.com/f170/e-herms-build-4500w-elements-pics-310259/
But now that I bought two pumps instead of one, I am planning on doing what you are talking about, and using the HLT as chilling vessel. I figure I can have one pump recirculating wort through the HEX and back to the BK. The other pump will recirculate the water in the HLT to keep it agitated. I’ll use water from the tap to bring the temp down from boiling quickly (maybe to 130-140) then transfer the (now heated) tapwater over to my (emptied and shop-vacced) MLT and add oxyclean to clean it. Then fill the HLT up with new tapwater and add ice to bring the wort down to pitching temps. Once the wort is at pitching temp, transfer to fermenter. Then transfer Oxyclean solution from MLT to BK to clean it. Use the second batch of chill water in the HLT as a rinse (starsan optional) through the MLT and then the BK.
At least this is how I picture it going in my head. There might need to be more than one batch of chill water, and it would probably be best to just keep the tapwater flowing constantly so I have the highest temp differential (maybe cycle the first 10 gallons into the MLT since it will be the hottest, then just dump or recirc the rest until I get down to 130-140).
One thing I think I will definitely need here is a hop spider. I can see things getting very ugly if the pumps/HEX gets clogged up.