Billing Rate Cards

You have a couple options based on some broad assumptions I'm making based on what you wrote but if it were me I would do one of the following: 

1. Have a 'current bill rate' that you summarize into the Project that gets snapshotted into the timesheet when it's created. This would just summarize the rate into the project from the current bill rate and then add as a lookup and snapshot on the timecard. This only works though IF the user enters a timecard that still falls in the billing period - so if someone is a day late in their time and the billing rate entry changes this falls off.
2. Pipeline when record is entered - lookup the matching billing entry for that timesheet and update the rate
3. Formula query that looks up the bill rate based on Project/Time period and pulls the bill rate, then on a daily basis you have a table import or pipeline that 'copies' the rate into a static field. once you've processed and copied the rate you nullify the formula query so that it stops processing and doing the query since it's not needed anymore.