Postscript from the original poster:
I went through THREE levels of supervisor, until I got a return call from, apparently, someone with authority. I nicely explained that I'd done everything that could be expected from my end, and asked her if it seemed reasonable for me to bear the burden of AMEX's and TWA's procedural error. She agreed that it didn't, and asked me to check my account again after the weekend, when she'd see if it could be fixed.
IT WAS FIXED.
As with so many of these things, perserverance is critical, but it's so sad that we have to endure this time and again.