How do I use bill rates from Activity Items instead of Employees?

If you want your time entries to use bill rates defined for the activity item rather than the employee, you can set that in the Projects screen using the rule Use bill rates and cost rates from activity before bill rates from resource. Check CORE Help for details on assigning rules to projects. Furthermore, you can set BQE CORE to automatically apply this rule to all new projects by selecting the option Use rates from activity list in Settings > Projects. To know more about the rate hierarchy in CORE, check out this article: Rates in CORE.