I went ahead and marked the answers by @ptoche as solutions, since they do indeed solve my original question. I would also mark my answer using ApproxFun as a solution, but I’m not sure what the etiquette is about marking your own answers as solutions.
Based on the discussion here, there’s definitely more to be said about solving these kinds of problems, especially as more state/control variables are added to the mix. I’m especially interested in seeing the pros/cons of using ApproxFun vs. EconPDEs after reformulating into a value function problem.