I am a *G (TG) and when flying economy class out of BKK on any airline, have NEVER been given Fast Track privileges, and that includes on TG itself.
Bottom line: if you are flying Y regardless of your status, do not expect any help from any airline on Fast Tracking you through immigration. Of course it's worth asking politely when you check-in, but don't expect it to happen. Plan your timing to complete immigration and security per worst case scenario.
If you are flying J/F, it depends on your airline policy and definitely worth asking about Fast track if they don't automatically offer up this option. If you are flying J/F on TG, head for your dedicated immigration and security at end of check-in Aisle A.