When I turn my iPhone 16 ProMax off overnight, TimeMachine backups complete without issue whilst screen is asleep. But when iPhone is on, backups fail. I think this a FindMy conflict issue. I have the FindMy widget on my desktop. Mac Mini M4 Pro.
That was certainly one of the exclusions I had to make for Time Machine to work when the computer is locked. One of the files which prevented a backup was in ~/Library/Containers/com.apple.findmy.FindMyWidgetItems/.
However, I don't have any FindMy widget anywhere on my Mac. It could be that using Find My on my iPhone when the computer is locked could trigger it.
For the last month, I've not had a single issue since adding the backup exclusions:
~/Library/Containers, ~/Library/Biome, ~/Library/Daemon Containers