I purchased an app via Google's play store about a year and half ago. Basically, I got a new phone, transferred data from my old phone to the new one. The app I'm referring to is there but as the "free version" without unlocked features.
The app isn't available in play store anymore, so I can't purchase it again.
Is there a way to transfer apk and data from the old phone, or am I out of luck?
There is an app called Helium, which sounds like what you're looking for. Supposedly, it moves app and app data between devices, which should include the recognition that you have the premium upgrade. It does, however, require a desktop to complete the transfer. There are other options like oandbackup and Migrate - Requires TWRP, depending on your level of experience and whether you can make certain modifications to both/either phone.
Yes, if you only just want to copy over an .apk, that's certainly possible and shouldn't require root (except for certain settings or user app data that may be saved to a system directory). The simplest method would be to use a file-browser that also includes an app management function. These are typically free, and simple to use (I use X-plore). With this app in particular, there is a function you can enable that will list all applications installed. Simply navigate on the opposite viewing pane to a device internal storage directory (e.g., /sdcard/documents/), then return to the app manager.
Long-press on the app you want and copy it (not move); this will copy over the .apk file (and automatically appends the version to the filename) and also avoid any permission errors. From there you can then share it to another device or external/network directory.
EDIT: You can also try searching F-Droid repository to see if the app might be there.
EDIT 2: Oh! Apologies, I wasn't careful enough with reading the OP, for some reason I thought you were referring to the free version. In that case, I'd suggest reaching out to the developer directly, as even if you do find a way to transfer it I don't think it would be wise to post such a method up, since others might abuse that knowledge. For this particular app, it's almost as if they don't want to be found! I've had some difficulty tracking it down (looked at the Play market archived page, then had a chat with Amazon and found this link as well). There's two separate e-mails for you to try, good luck! Oh, and here's a Twitter account.
