Since they just look at the DVC-Y on the phone and don't actually scan anything, I wonder if you could print a photo of the card page on your phone? On cruises, I bring along a laminated copy of my DL and passport for boarding the ship at ports of call, because they always ask for those, yet I don't want to be hauling my passport and ID all over various islands and countries if I don't need it for anything except getting on the ship. Those laminated copies work just fine along with my cruise ID/keycard.
Would Disney honor a copy of that page, laminated like a card, at a place like a waterpark where someone may logically not want to have their phone on them?
*different subject, but I keep a copy of my driver's license on my iphone, and use that at the DVC lounges when they ask to see my ID along with the DVC-Y. Since I'm already showing them the DVC-Y on the phone, it seemed easiest to also have a clean copy of my ID there so I don't have to bring my wallet or ID to the parks. I have had an occasional cast member say they didn't think they were supposed to allow it, but they do anyway - and most (90%) don't even bring it up. Some states have actual verified digital license cards, but Florida is in-between systems now and doesn't have one...so this has worked in the meantime.