Well, the good news is that I don't think you're doing anything wrong and there does seem to be a way to continue authoring your project on iPad.
I had to ask a friend with an iPhone to test this for me, so I hope this works the same for you. Safari on iOS doesn't seem to want to open .deck files at all, so we'll use the other file format that Decker projects can be saved to -- HTML.
1. Save your project as a .html file instead of a .deck file
If you're not sure how to do that: in the Save As menu, write .html at the end of the file name.
This will have a slightly larger file size than the same project as a .deck but that's fine and normal.
2. When you want to open your project again go to a copy of Web Decker hosted online.
For example: The Decker Tour Deck is online, unlocked, and always there for you.
(Yes, it does seem like you have to do this instead of opening the .html project file from your device's File system.)
3. Use the File > Open menu inside of that online instance of Web Decker to open your project.
Then make sure save as .html again when you're done!
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Your existing .deck file should be able to be opened in Web Decker on a non-iOS device and saved as an .html file so you can continue using it on iPad with this method.
Web Decker is nearly identical to Native Decker when it comes to authoring projects. One of the main differences between them, unfortunately, is that there's also a web browser involved in Web Decker. And every web browser has it's own slightly frustrating quirks. I'm sorry you've run into this one.
Anyway! If the method above works for you, I'd love to know.