Possible to re-add discontinued app (.ipa) to iTunes?

This requires some introductory info:

Many months ago, an iPad app that I use regularly was removed from the App Store. It continues to function perfectly well, but yesterday, it had trouble connecting to its servers.

Last week, after installing iOS 7, I'd had a similar problem with the YouTube app: It would launch, but never connect. I solved that problem by deleting and re-downloading the YouTube app. So, yesterday, I did the same with the aforementioned app -- forgetting that it had been removed from the App Store and couldn't be downloaded again.

The developer of said app also has an iphone version, which I run on my iphone. But they've never replaced the iPad app that was removed from the store, which is why I've been running the old version.

Finally, in an attempt to manage the 1,100+ apps that sit in iTunes, I routinely go through and delete apps I don't need (or don't think I need). Since the icons of the aforementioned iPhone and (discontinued) iPad app were identical, at some point I deleted the iPad app from iTunes, not remembering, at the time. my reason for keeping the older one.

The good news:

Using Time Machine, I was able to roll back, retrieve the deleted iPad app, and restore it to iTunes' Mobile Apps Folder.

The bad news:

I thought restarting iTunes would cause it to "see" the previously purchased app that was reintroduced to the folder, but it does not. It appears that once you delete an app from iTunes (and move it to the trash, versus choosing the "keep in my library" option), iTunes completely severs all ties to that app -- even though you purchased it legitimately. I thought apps would be "discovered" at launch, just as songs you add to the iTunes folder are "discovered."

Short of using Time Machine to replace the entire iTunes folder (which would be an epic nightmare) is there any way to get iTunes to "see" the app I retrieved? Is there one document, perhaps, that iTunes uses as it's app "registry" that I could restore via Time Machine, in order to "bless" that app again? Of course by rolling that document back, I'd be rolling back to a point that's *before* current apps were purchased. Many, many current apps. So, this is quite a conundrum. But I really need that app back, so I'm willing to try anything. I'm running iTunes 11.1.

Suggestions?