I use IndexedDB for an Anvil app that primarily has IOS users needing offline access and things seem to work fine with it (of course, this depends a lot on which IOS version they are using). Only issue is that there is a storage limit of about 1GB which seems fine for my use case.
Another thing you can do is to build your own native app that shows your anvil app in a webview. Then you can do communications between your native and anvil app (and let the native app handle storing of data).
It’s more work than a converter but you’ll have more control of the app. Here’s an example I did with react-native