Hi Beniamino,
I'm afraid I don't have a clue about why that one screen only works on one of the phones, but perhaps you're making more work for yourself than you need to?
I'm referring specifically to your comment about not wanting to reinstall and reconfigure everything. Having been supporting the iPhones the entire rest of my family uses, I have often had to swap phones because they've upgraded, or they need a temporary phone while I repair theirs, or they've broken it beyond repair and need a replacement. In any case, the most straight-forward way I've found is to do a full backup onto a laptop, making sure to use a password. When you do that, it can backup everything, especially the keychain containing your passwords and other secure information.
Whenever I do that, then restore that backup to the replacement phone, the new one comes up looking exactly like the original, with passwords, passcodes and all other configuration the same.
So while I don't know what to do about the screen, it seems you can just use the working screen on phone B then backup phone A and restore it to phone B. You won't see any difference once you've done that.