Time machine backups store the files saved on a particular date together in a folder of that date. Other files present on that backup date are represented only as links, to files stored elsewhere in the backup set. Searching through the sets of what was saved on a particular day for a particular file is an exercise in futility.
TimeMachine.app (not to be confused with Time machine preferences) is the included Application that allows you to see into the Time Machine Database, and shows you re-constructed versions of folder as of any given date still on the backup drive. once you find a folder with a given file of set of files, you can ask Time machine to retire that set of files to a location of your choosing.

This would be MUCH easier using the computer and its backup together, rather than trying to get into backups for one computer from a different computer.