How can I use Global Entry lanes at Canadian airports if I don't have the physical Global Entry card?

When you are approved for Global Entry, U.S. citizens, U.S. lawful permanent residents and Mexican nationals are posted a Global Entry card. Everyone else only has an electronic record of their Global Entry status.

Recently, a number of additional countries have become eligible to jump through all the hoops + pay to apply for Global Entry, the UK amongst them. This applies to myself.

Earlier I passed through the US pre-clearance facility at Vancouver YVR airport. That featured a separate security line that then fed into the Global Entry and Nexus machines, which was signposted as being for Global Entry holders, Nexus holders and diplomats.

Unfortunately, because I don't have a Global Entry card, they wouldn't let me use it. They seemed unable to comprehend of how someone could have Global Entry but no card, and kept saying "but you must show us the card you were issued or the sticker". I offered to show them the status online, but they wouldn't go for it, so I was forced to make the walk of shame back to the end of the normal security lane, then much later duck under a barrier to get over to the Global Entry machines.

Should I have done something different? Is there something I could've shown the Canadian staff manning the security checkpoint that would've let me use the Global Entry line as someone who genuinely has it and is entitled to use it? Printout from the DOES site perhaps? Calling the site up on my phone? Anything else?