Mac – Apple ID or password is incorrect when restoring purchases from another Apple ID

I am having issues with trying to use software I purchased from the Mac App Store. I have two Apple IDs: one personal Apple ID that I've been using for years (an iCloud account) and a new Apple ID that was created when I set up my company-provided Macbook. The "company" Apple ID is associated with my work email, and is the primary Apple ID for my OS X user on my work laptop.

I was using software on the work laptop which I had purchased under my personal Apple ID (because I had paid for it myself; my work Apple ID does not even have a payment method associated with it) and everything was fine until the App Store updated this application.

After the automatic update, the application was back in "free trial" mode. When I clicked "Restore Purchases" I was asked to sign in:

Sign in to download from the App Store
Enter the password for your Apple ID.

However after signing in with my personal Apple ID I was told the password was incorrect. I'm 100% sure the password was correct.

Your Apple ID or password is incorrect

What's really odd is that *after I click Cancel_ I get a second dialog asking me t sign in, but this time it recognizes that I need to use a second Apple ID.

Sign in to use this application on this computer
"App name" was purchased using the app store on another computer.
To use "app name" on this computer, sign in using the Apple ID and
password used to purchase it.

But even after signing in (and yes, I am sure the Apple ID and password were typed correctly!) I still get (in red)

Your Apple ID or password is incorrect

I am using Mac OS X 10.15.2 Catalina. I have signed in and out of the App store using both Apple IDs and even added my personal Apple ID as an iCloud account on the work MacBook, but nothing seems to resolve this issue.