How do I make the user's 'library' folder within a Time Machine backup available to 3rd party apps?
As the latest backup is always going to be in the same place, then it's going to be the same path as ~/Library, plus the TM path.

Regular path is ~/Library/Application Support/MobileSync
or longhand Volumes/[boot drive]/Users/[user name]/Library/Application Support/MobileSync
so in Time Machine it will be
Volumes/[TM drive]/Backups.backupdb/[mac name]/Latest/[boot drive]/Users/[user name]/Library/Application Support/MobileSync

Your larger problem may be that the folders in there have special permissions rwx------ &amp; you cannot get any further than Library using Go to Folder... from the Go menu.