Dare I say, it might be unnecessary. Being on the Gulf Coast is like being in one of my totes. When I lived in SE Houston, I rooted my cuttings in containers outdoors. About a year ago, we vacationed in Galveston. I took the trees I was propagating with me (don't judge me. We're all crazies here) and set some rooted cuttings in Air-pots on the deck rail of the beach house. The roots which came out of the holes extended over half an inch into the salty, humid air without the tips dying back.
The concern I have with your set up is the potential for the cube to become saturated with water and rot the cutting. I use a spray bottle to mist the cubes just until they are damp. If the cups you have fit snugly in the tote, you could use the empty cups to hold the cuttings/cubes until you see roots poking out, then add your growing mix.