It took some googling to figure out, but eventually I came across the answer to this problem across a combination of a couple of old forum posts. I'm writing it here on the off chance someone else needs this. Or if I need it again.

• Log into the IP Office R5 Manager
• Navigate to the Short Code tree
• Find the ? entry - the feature name is Dial
• Change the Telephone Number field to one of the following depending on what you want
  • . - this will withold the number you're dialing from
  • .S<number> (eg: .S01234567890) - this will show <number> when you dial

<number> should be your number. If I set it to a number I don't own, it just shows the one I'm calling from.

Sometimes a business has some system or piece of infrastructure that just works. This is a good thing! Unfortunately, some businesses don't like to invest in technology unless that technology breaks.

In related news, I recently had to figure out how to toggle a setting on some old software, Avaya IP Office R5. As someone who is totally not a phone/VOIP person this was a challenge. Add on to that the fact that the software isn't very well designed and you have me fumbling about trying to understand terminology and remember where obscure settings are just to do the basics.

This software was managing a 15 year old phone system, and since its inception had been showing as a withheld/private number when calling out to external phones. Not sure why this was ever set, but it's off now.