Messages won’t sync to iCloud or transfer to new iPhone

Here’s our situation: my daughter is trying to migrate from an old iPhone 7 to a (less-old) iPhone 12. The entire migration went without a hitch except for one giant-sized hitch: roughly half of her Message history does not show up on the new phone. On further investigation, we have determined the following:

• iCloud Message syncing is turned On on the older phone. However, on that phone Messages are taking up 5.6 GB of storage, and in iCloud messages are only taking up 2.8 GB of storage.
• On her Macbook Air (which also is signed in to the same iCloud account) Messages are taking up 2.8 GB of storage (same as in iCloud); and lots of the messages that she can see and read in her old phone do not show up in her Macbook Air. These missing messages appear to be scattered randomly throughout her message history (or at least we have not been able to find any pattern to which ones are missing), and (as far as we can tell) are the same messages that are not showing up in the iPhone 12.

All of this seems to strongly suggest that the problem is that messages are not uploading from the old phone to iCloud. The most obvious thing to try next would be to turn off iCloud Message syncing on the iPhone 7 and then turn it back on again to see if that forces a resync. However, when we attempt to do that we get the “Disable and download?” prompt, and we are very leery of going down that path, because (shocking twist) we have been in this exact same situation before, with a different (even older) phone, and when we hit the “Disable and download?” button all of the messages that had been locally stored but not backed up to iCloud were permanently deleted from the phone and could not be recovered.

(Parenthetically: I don’t know if this is what “Disable and download?” is supposed to do, but we have also encountered other people online who have had the same experience, including some who have explicitly warned users not to trust Apple Store “Genius” employees, many of whom will tap that button without understanding that it has exactly the opposite effect of what is intended.)

Just to complete the story, we have tried setting up the iPhone 12 three different ways:

• From an iCloud backup — this obviously did not work, since the iCloud backup is incomplete.
• From a local backup on a computer. She backs up her phone regularly but apparently (and to my great frustration) those local backups don’t contain her full Message history either, presumably because she has iCloud syncing turned on.
• Using “Quick Start”. This seemed like a promising idea — skip the iCloud middleman and transfer directly from one phone to the other! But when we opened the Messages app for the first time on the new phone it was initially empty, and started downloading messages from the server.

So here is the ask:

• Is there some way to force the old phone to resume syncing without tapping the dreaded ‘Disable and download’ button and risk losing all of her messages?
• Alternatively, is there some way to force a local backup of the phone that contains her message library, again without tapping the D & D button?
• (Getting increasingly desperate:) Is there some way (including, I don’t know, a command-line tool or 3rd party app) to export her full Message library from the old phone and import them manually into the new phone?
• If neither of those can be done, is there some way to export the entire Message library into some giant text archive or PDF file so that she can at least keep and read her old messages, even if outside of the Messages app itself?

Any advice you can offer would be appreciated, we are at our wits’ end with this. Practically her entire teenage social life was conducted inside Messages and she really does not want to suddenly lose half of it at one go.