All,
Since I've now managed to get through this process, I thought I'd share a little about the experience and how it works. This is how it worked for me at a flying club, but it's much the same for the airlines/corporate gigs.
Information dated May 2016.
1) Find a sponsor for your flying (airline, club, company, etc).
2) Apply through them for validation on your ICAO license. They will sponsor you to the Civil Aviation Authority of Thailand (CAAT, formerly DCAA) for the written test. You must submit the requisite paperwork - copies of certificate, medical, passport, etc, plus a fee somewhere around 2,000 THB (60 USD) to CAAT.
3) They will process all the above, and about two months later, you will be scheduled for the validation test on Thai Air Law. This 20-multiple-choice-question test is given in Bangkok on a computer, makes almost no logical sense whatsoever, and is pretty easy if you cram from a good study guide (PM me for details). The material has next to nothing to do with actually flying in Thailand. Example: What year was the Thai Air Navigation Act passed?
4) Bring the test results form back to your sponsor, and submit through them an application for validation, with more copies of all of the above paperwork. Wait another month for your existing certificate to be verified with its governing civil aviation authority.
5) Receive a license to fly in Thailand... sort of. Your license is good ONLY for the specific aircraft operated by your sponsor. Could be every tail number in an airline fleet, or just the one or two planes at your company/club. Your license will have only enough privileges to cover the aircraft and operations you'll be flying - if it's a single-engine, your license won't include multiengine. Not an IFR-certificated aircraft? No instrument rating. Flying at a club? Your ATP is now a private pilot license. (If you've bought your own aircraft in Thailand, your license will be good only for that specific tail number. Rich enough to buy a second plane? Great - you'll need to apply for an additional license, specific to that aircraft.)
6) Fly.
This is why there aren't many foreigners flying in Thailand... But it can be done, with enough persistence and patience.
Good luck, and let me know if you need some help navigating this insane system.