Ok. Phones supported, no issues there. So with built-in bridge, you are actually using the DSP within the phone that you is being barged (the target). In your scenario, this would be your agent. My first questions then would be:
1) Has Barge ever worked?
2) Have you tested cBarge to see if it behaves as expected?
Barge issues typically boil down to configuration issues. I tend to set things at the cluster level and avoid phone-level configuration issues (or potential thereof). So, Enterprise level set Built-In Bridge to True and Privacy to Off. I normally set the party entrance tone to True as well. Then you just need to assign the Barge softkey to a template that is applied to the phones. No phone-level configurations unless someone wants Privacy. The message you are receiving basically means that there are no DSP resources available on the target phone to set up the conference call. Your settings sound correct, but since it is being done at the phone-level I would at a minimum double check at least 2 test phones, verify configs, and reset the phones...then test the operation.
In 6.1, there is the possibility of a bug as well but I haven't looked at this specifically.
I'll see if I can think of some other possibilities as well.
