I decided to try leaving the cabinet in the sun to heat it up, and things went marginally better, but considering the amount of vinyl I figured a minimum of 8 hours or so for the heat and peel method.
So I grabbed the sander and gave it a go. Things went great! I used a random orbit sander and 80 grit paper. It took a couple hours, and running the sander was much more enjoyable than blistering my fingers peeling it off.
The vinyl on the cabinet appeared to be the same stuff the Happ sells on their web site.