Shopify support again said they're aware of the issue but gave no indication when it will be fixed.
Deleting the Shopify app each morning, reinstalling and re-launching the store allows us to print - we've been doing this every morning for weeks.
I wish Shopify would assign top priority to these sorts of issues, it's not brilliant but obviously it's acceptable to Shopify.
It sounds like the latest app version fixes this issue, although app updates theoretically shouldn't break things in the first place. If there are properly tested before releasing on the App Store.
Our workaround usually consists of power cycling all the equipment, and if there still is an issue we reset network settings on the iPad. That tends to get things working again. There are other fixes listed here, which work equally as well.
It would be nice if Shopify notified their users via e-mail of large-scale issues that are reported into their support staff. If it wasn't for resources like this message board I wonder what customers would do?