Now my thought was to make a short adaptor cable with an Edison plug on one end that ties to only one leg (black, green, white in this case) of an L14-20R. I know this means I can only use one half of the quad but I can live with that.
In the female end of your adapter,
if the terminals provide space for it*, you could run a jumper between the black and the red slots, then you'll be able to use both halves of the quad box. It will effectively make it as though there was no red wire, and the hot sides of the receptacles were jumpered just like the neutral sides (white wire). Think of it this way: instead of the jumper being in the quad box, it's in the female end of the adapter. The effect is the same.
*NOTE: You can only do the jumper if the terminal is either the kind that allows you to wrap the wire around the screw (a method not advised with stranded wire) or where you can insert a wire between plates on each side of the screw. The jumper should be the same gauge and stranding as the main conductor. Many connectors only allow a single wire inserted between the plates that are clamped together by the screw; there isn't a good way to jumper with this style. Under no circumstances should you split the strands of a single conductor between two terminals, as that could result in an overloaded conductor between the split and the terminals.