In the end, I used joint compound to build out the gap.
I used wood shims to create a backing and set a depth for the fill. To avoid compound sticking to the wood and to also give a small shadow gap, I added strips of cardboard wrapped in plastic from a cut up letter file. Next, the 3/8" gap was filled with joint compound and left to dry for 24+ hours. Careful sanding allowed me to merge the fill with the existing wall. A bevel was created by even more gentle sanding after the protective pieces were removed.
Here is a photo just after surface sanding showing my setup:
After a good lick of primer and paint, a capping wooden piece is installed on top of the half wall. The joint compound fill looks just like the rest of the wall. The result is good enough for my purposes!
Just be sure to let the joint compound dry fully before sanding (dry time may be extended because of the thickness).