Please bear in mind that it has been many years since I worked with WP8.1 and I have forgotten some details. However, I'll do my best.
Also, be aware that it's entirely possible that the data you seek is gone forever. Neither of these approaches will give you the ability to magically "undelete" files (you can try using a file recovery tool if you do the USB Mass Storage route, but if the files were deleted some time ago they've almost certainly been overwritten).
Finally, note that these instructions assume the app in question was installed to the phone's internal storage, not to the SD card. There are variations on the instructions that might work if the app is on the SD card, though that's actually harder to recover in some cases (the app data on the SD card is encrypted; not hard to defeat the encryption but it can complicate recovery).
The SD-based interop unlock worked by moving some OEM app from the store (one that had ID_CAP_INTEROPSERVICES) to the SD card, then using a sideloaded app to hijack the app on the SD card (there's a process there, but it was mostly automated by the end, just install and run the relevant apps). That gives access to the hijacked app's capabilities.
A full tutorial for the SD-based approach is here: https://forum.xda-developers.com/windows-phone-8/development/interop-unlock-oem-unlock-windows-phone-t3467191. Note that this hijacks the "MixRadio" application, which is pre-installed on Lumia devices but was discontinued long ago; if you uninstalled it at any point than some modifications are needed. The tutorial says to hard-reset as the first step (so you get the over-privileged MixRadio app back) but that's obviously not viable for you. Here's a slightly older tutorial that uses Nokia's "Extras + Info" app (normally present in Settings IIRC, but technically an app): https://forum.xda-developers.com/windows-phone-8/development/tut-interop-unlock-root-wp8-8-1-windows-t3450239. That one doesn't require any hard resets but is a bit more convoluted. If neither of those work I can try walking you through the process for manually hijacking an app chamber, but it has been many years since I did that and I don't have a WP8.1 device to test on; I'd be going off my old tools and forum posts.
Alternatively, you can use the WPInternals tool, which allows unlocking the bootloader of the phone. Note that if the phone's file system is encrypted (BitLocker), this will not do you any good!
In the tool's earlier days you often had to re-flash the entire OS image (which of course erased all user data), but these days it is in theory possible to overwrite only the portion necessary to allow booting the phone to USB Mass Storage mode, after which you can connect via USB and browse it like an external hard disk with full access to all data. I must stress IN THEORY because this is very "hacky" and does not always work perfectly, and there's a risk of it putting your phone in a state where the only recovery option is to reflash it. That gets you a working phone again, but at the cost of all your data.
The tool is introduced in this post: https://forum.xda-developers.com/windows-phone-8/development/windows-phone-internals-unlock-t3257483. However, note that that's a VERY old post. These days, unlocking the bootloader and enabling USB Mass Storage mode is quite easy on supported devices, assuming the tool works correctly. The download is at https://www.wpinternals.net/index.php/downloads,
Assuming you successfully get the phone unlocked and into USB Mass Storage mode, it becomes possible to browse the file system; the data you're looking for should be in a folder on the Data partition. The path will be something like \Users\DefApps\AppData\Packages<IDENTIFIER_STRING>\Local, although I almost certainly have some part of that path wrong. Poke around until you find it. DO NOT try to write to the phone's file system (including stuff like opening a file and saving changes, renaming a file or directory, or editing the access to a directory). The identifier string will either be a GUID (if the app is old), or a text string that probably contains something related to the app name followed by random characters (for example, the ID string for the Windows Mail / Outlook app is "microsoft.windowscommunicationsapps_8wekyb3d8bbwe"). Figuring out the right app - especially if it's an old one with a GUID instead of an ID string - may take a lot of guesswork.
In either case, it might be hard to tell what you're looking for. If the app stores its data in simple files (plain text or a common format like a JPEG image or something), those will be easiest to recover but are the least likely for the app to have lost them. If the data are stored within a structured file system hierarchy without clear names, it's probably possible to find them but might take a lot of looking. If they're stored in an archive (such as a ZIP file), you won't be able to find the data themselves just by looking in the folder but you might find the archive and open it with something like 7-Zip. If they're stored in a database (such as SQLite), then you'd need to extract the DB file and use another program to pull the data from it; this can be complicated.