I'm sorry if I muddied the waters by suggesting a manual setup as a "New iPhone" might be easier than a restore and setup with USB from the Finder (or iTunes depending on your macOS). Obviously this has worked well for me and others where their macOS is earlier than the current version or they just want to start afresh with the bare bones, a little like a "clean install" of macOS. I had no idea that you were getting an iPhone preset to a carrier and obviously this was part of the problem.
It occurs to me that it should be fairly simple to change one the of the Apple ID's so that they match and simply ignore the other but I'm reluctant to suggest you do that on your iPhone so that means changing it on your computer.
I'm not sure what others think about that because when you say, " Generally, I have managed to stay away from iCloud, iTunes, App Store. Don't use 'em, don't need 'em." It might surprise you to realise that all Apple devices use iCloud and the App Store for a number of things. Mostly it is to keep all the devices in sync and provide data security and backup throughout.
Below is a picture of some of the data that is stored and shared (synced) on my computer and it is the same on all my devices. eg. If I add a contact, calendar event, a Note, etc on my computer it appears on my iPhone and vice versa.
It's likely your computer and other devices are the same or you would be receiving prompts (notifications) to sign into iCloud on your computer and iPhone as have other who attempted to avoid any online contact with Apple. MS Windows is the same.
Lets see what Jake or others say about this first, but I would check your contacts and data on both devices and if you find that all your data is complete on your iPhone consider changing your computer to the same Apple ID as your phone. This may mean you would have two of some things, such as you Apple email address but that
shouldn't matter.
I dont want to make things too complicated for you but I do feel that having two Apple ID's, one on your iPhone and another on your computer is going to cause all sorts of problems into the future especially with 2FA which we might discuss at a later time.