This iCloud backup includes purchases that were made with a different apple ID

... is the error message I keep getting when setting up my new iPad. I changed my primary email address several months back when my provider ceased offering email services, which was also my Apple ID. However everything seemed to be motoring along okay, until now. I can’t remember the old password, although I’ve tried all the most likely suspects. When I try to use Forgot password, it tells me that the old Apple ID does not exist, and it just wants to change the password on my current Apple account, which is not the problem. Any ideas? I didn’t get a new Id, just changed the name of the old one.

Context: There is some old game data that I’d like to bring across (on a game purchased under the old ID), which cannot be accessed any other way, which is why I’d like to use a backup.