Over $6000 to install TTI headers on 66 Barracuda

wow, I have been blasted on this thread. Some of it is deserved so I am not offended.
Not by all of us, James.
You have your own reasons, as you stated. This should have been a quick, 1-day in & out.
I suspect your mechanic is a pimple-faced kid who works on ricers all day and your 'cuda was built when his parents were dating...or before.
This doesn't make him a BAD mechanic, necessarily...it makes him INCOMPETENT to work on your classic Mopar.
Maybe he can tweak a buzz-bomb just fine?
Anyway...NOT YOUR PROBLEM.
Pay him what the original estimate was (along with a bag of rice), collect your car and giddyap...take one of the guys here who offered to help up on their offer. And in the future, be prepared to learn your machine. It's the only way to guarantee work done properly with no shortcuts.
Good luck, and let us know how it ends.