Installing old app that's no longer on App Store

Hello - I recently got a new iPad running iOS 13. On my old iPad (which I still have) I have an old version of an app that is no longer on the App Store but would like to keep using. The app still works on the old iPad.

I've set up the new iPad using a backup of the old iPad, and it restored just fine. The only problem is the new iPad still makes me download all my apps again (the icons are there - you just have to tap each of them). This is easy enough to do, but it downloads the new version of the app I want, rather than the old one.

I have a backup of the ".ipa" file of the old app on my Mac, and in the past I think it was possible to add the app back using the "On My Device" -> "Apps" menu in iTunes, though this functionality no longer exists. But I believe it's possible to drag the ".ipa" file to the "On My Device" section. The only problem is that when I do this iTunes says that my Mac needs to be authorized for my iTunes account, but it already is.

Do you have any other suggestions for putting the ".ipa" file on my new iPad? Thanks