Reinstalling an app that was removed from the App Store

I recently upgraded my iPhone and restored the backup of my old phone onto the new one. One app showed a cloud-and-arrow next to its icon. If I tap the icon, it will try and install the app but comes back with a message that the app is no longer available in the App Store, and fail.

I have the app's .ipa file in my iTunes folder. I have tried to install the app by dragging the .ipa file into iTunes with the phone connected, but I get a message that iTunes cannot install the app "because it has already been installed".

The obvious solution would be to delete the app from the phone and reinstall from iTunes, however I definitely want to keep the user data associated with the app. If I delete the app, I lose that. Is there a way to force iTunes to install the app and retain my data?