Is there a way to bypass 'traditional' Time Approvers in the Time Entry process to allow for an attribute on the Project Object to be the approver? For example, we have a need at my customer to use the Project Director instead of the Project Manager or Resource Manager. Because the OOTB is heavily dependent on these 2 fields, I cannot see a way to use any other field, other than the Project Manager or Resource Manager.
Help and direction is much appreciated.
If you look at the stock "Project Manager Approval" process for approving timesheets you should find that it does the approval based on action items directed to different project managers (rather than time-approval access rights).
So you could build something similar, relying on Action Items directed to your "Project Director" to approve a timesheet and this could implement your required timesheet approval that way.
Building on David's comment: In order to route the action items to your Project Directors, you'll need to create a new attribute on the Project object that uses a 'resource' based lookup. Using a resource based lookup will result in your new "Project Director" attribute showing up in your workflow process as one of the Resources that one can route an action item two.
Here's a link to how to create such an attribute: http://www.ca.com/us/services-support/ca-support/ca-support-online/knowledge-base-articles.tec438924.html
One can use an OTB resource lookup or create a new one - the link provides instructions for both.