I think that it works, but for it to work you need to enable Google Play services. From what I understand, this is done in a sandboxed manner simulating a fake identity, so it is possible to do this while isolating Google from your phone to an extent. But I think that WhatsApp is in itself problematic and one of the direct offenders that I want to avoid, regardless of its reliance on Google Play services, and so I have not gone through this effort myself.
I am not sure as I did not test this one. Maybe you can go in person and get a worker to get you access to the kiosk through your account to print the card. It is one of those massive chains with gyms in every corner. I think that by now they rely on their digital infrastructure and many of their workers are not trained to handle uncommon situations. At least I get that from some of my experiences, but I could be wrong, maybe if I would have called them could have helped me with this. It was just easier to get the app into my old phone, print a card, delete the app.