OTL/Time and Labor: How to link or automatically populate fields on timecard

Summary

How do I link Objective field on timecard to populate based on selected Cost Centre?

Content

Hoping someone can please help me with this requirement on timecards:

We currently have a field configured on the timecard called Cost Centre Override that allows employees to choose a Cost Centre if they are not booking their overtime to their default Cost Centre.

There is a new requirement to also have the Objective associated with the chosen Cost Centre come through as well. It is preferable for the Objective to be automatically populated, rather than allowing the employee to select it from the list of values - this will avoid any mistakes.