Tom,
Putting a QE under a hive is a short term emergency measure only.  The bees
won't like it and neither will you.  Drones will be caught by the neck and
die.  A drone escape is also a queen escape.  Some years ago a chap called
Paul Mann had an idea of using a QE in a special floor board to divert a
departing queen and therefore the swarm into an empty hive alongside the
potential swarmer.  I made up such a board but never used it so don't know
whether it works.  If you're interested I may be able to probe the inner
recesses of my memory for details.
I suggest the best way forward for you is to make artificial swarms.  If you
don't want to increase your stocks on site put the "swarm" back on the floor
with the parent colony containing the brood and any queen cells above it
separated by a "split board".  Re-unite them when the new queen is laying or
run them as a 2 queen system for a while.  Reduce to 1 box at the end of the
season.
Chris Slade