I think the agent you describe, who would streamline/handle everything, is more the "pet transport companies". (You arrange everything with them prior to starting the process, and they hold your hand/do things for you throughout the process, not necessarily just the airfare part.) If I recall correctly, when I very casually looked into it, the cost was ~$2,000 (for 2 pets), so getting it for $650 is unlikely.
Having to pay for the company to make a call may just be the cost of doing business. I didn't have an option to not use an agent at Heathrow/ARC for my pets. Since it was included in the price, it was just a matter of faxing them a form, so it wasn't a big deal, but with Virgin Atlantic you have to pay extra for it, and it's not optional.
$530 for transport sounds pretty cheap. I don't know how big your dog is, but I paid ~$2,300 to ship my 60lb dog and 13lb cat over, and I think the cat was about $500 of that. If your dog is > 15lbs, I would assume that the transport cost/quote should have included the $650 charge, and there was just miscommunication about when/where/how it would be paid.
Good luck with getting help from Delta. Posting on their Facebook page (do they also have Twitter?) could speed things along, if they monitor that sort of thing.