If anyone else has this problem, I figured out the issue. It turns out that the programming I described was correct. The feature won't work until a call has been established for a certain length of time.

I would make a test call, then try the service code from another deskset and get an error tone. If I waited a minute before testing, then it worked just fine.

Unfortuantely we've been left to fly on our own with this system, and NEC has been deliberately unhelpful. It's been an interesting challenge.