Of course, right after posting I figured it out: the WiFi was interfering. Disabling WiFi on the phone made the stuttering go away entirely. Fortunately, I already had an app (LLAMA) which can automatically disable WiFi in car mode, which is exactly what I need. I do hear some interesting artifacts when listening to classical music, but I chalk that up to not having aptX on the phone. I just need to make sure my next phone has it.

Thanks for the offer of support. Good to know the Outlaws are there to help when we need them.