A software update is required to connect to iPhone

I know, this comes up repeatedly, but...

I’ve just updated my iPhone 12 Pro to iOS 16, and my iMac to macOS 12.6. I normally connect the two via USB to backup. Now when I try that I get a pop-up saying “a software update is required to connect to iPhone”; when I go for “Install” it downloads the software, then fails to install. However, my iPhone still appears to backup successfully despite all that. Both phone and iMac are running the latest OS.