Provide a user with service start/stop permissions
For posterity... I have no idea what I was doing wrong before, but I revisited this and was able to get it to work twice (a second time just to make sure I wasn't crazy) using the exact process I described above with a group policy object.
The only thing I can think of that I might have done differently was giving the Backup user Read privileges as well as Start/Stop/Restart privileges on the services.
Thanks to those who helped.