When you reach their corporate security rep, before going into any detail about the situation that led to your ban ask them if they can provide you with all documentation, comments, etc. that led to the ban. Hear them out as to their specific reasons. Then ask them what the process for appeal is. Don't elaborate anything on the phone. Anything you say won't likely be relevant and it may be recorded so you want to choose every word carefully. You need the opportunity to put in writing (succinctly and leaving out any hyperbole) your appeal and just hope for the best. Sorry, but if I were in the gate area and saw an apparently irate passenger attempt to open the gate door, I'd have no sympathy for them. And if your history of confrontational behavior with gate agents (as you eluded to in your first post) is on record anywhere, that will weigh negatively on your appeal. We've all had issues with power tripping, ill informed or just plain unhelpful ground staff.

Is your dog small enough to fit under a standard coach or domestic F seat if you're going to be relegated to another airline?