I think that using short storage totes would be your best bet, either as a drip or
ebb and flow system. Taking one and making a bunch of small holes in it that are too small for the
growing medium to go through. If using perelite or vermiculite as the growing medium, laying down a screen at the bottom. Then placing it inside the other. If using it as a
drip system you would only need one return line out the bottom of the second one. An ebb and flow system would be a little more difficult to set up with two lines through the bottom, and setting the overflow level but would also work.
By using 2 containers one inside the other you will be able to keep the bottom of the roots from sitting in the inevitable small amount of water at the bottom, and you will be able to lift it out and trim them when they want to go through the holes. That will probably give you enough root space, and the low clearance you need for that growing space. I got the storage totes with lids from Walmart a while back for about $6 each. What do you think, would that work for you?