I have fought so many insurance denials with a crappy small insurance carrier we used to have (it happens way too often) and I usually got what I wanted in the end. It's time-consuming and a very frustrating process because you get the run-around, but if you persist you hopefully won't have to pay more than the $1,000 you expected. When you call, ask to speak to a manager, or someone who can actually make a decision. Document your phone call in a log--date and the person you spoke with and what they said they will do. It will help hold individuals accountable and make things happen. Tell them what you want to be done (you may have to be little pushy and ask to speak to someone above them if they can't help you. If they say it can't be done, don't let them get off that easy. Ask them what they CAN do to help you), confirm what they will do and when it will be done by at the end of the phone call, and make sure they agree to call you when it has been done. Most likely you will have to follow up and call them instead of the other way around, but keep on it. Obtain their direct phone line/email if possible so you're dealing with the same person and they have more accountability. The squeaky wheel gets the oil. But be prepared for things to be dragged out. The appeals process, or any process with an insurance company for that matter, takes FOREVER with so many exacting requirements and steps.
Try to be respectful but be very firm. Don't buy their stories or runaround. Call them on it if they are changing their story, and refer to your notes. Let them know you are documenting everything, and "according to my notes when I spoke with you on such and such a date, we agreed you would do x. It hasn't happened yet, so I would like to know why, and when I can expect it to be taken care of." If you attack them (it can be so frustrating that you want to yell or cuss or spit nails) it will not motivate them to help you.
Start the whole process by telling them you have documentation that says patient pays "0%" for Dr after $1000 deductible and that you insist on them covering it after the deductible or that they give you an explanation.
Also, is the coverage handbook you are referring to current? I remember once I was looking at an outdated handbook, so my coverage had changed and I wasn't aware of that. So that unfortunately may be the issue.