The management interface is strictly for managing the device. The management interface will not pass traffic destined for anywhere other than itself. There is no "fix" for this. That would be a major vulnerability if traffic were allowed to pass to ...
The delay is not in the failover. The delay is in the traffic flowing through the 6513's now take a different path. I assume you are trunking your 6513's together, and thus that's how you're dual-homing devices to your 6500's and connecting them to ...
Since your 6513's are not in a VSS you cannot etherchannel/port-channel a firewall (or any device) split between your two chassis. The only way you can do this in the non-nexus line is to have them in a VSS. Your best bet is to etherchannel firewall...