[Glitch]

Here's my cheapo doomgaze strat:

Get your highest level, strongest caster (probably Terra or Celes). Teach Fire 3 or Pearl and equip a safety bit.
When you encounter doomgaze, he casts a spell that instantly kills your entire party. Safety bit will keep at least your caster alive. Immediately cast fire 3 (and if you're brave, go for a second) then RUN AWAY and sleep at Gau's dad's house
Doomgaze keeps his current hp between encounters, so you can just whittle him down like that and eventually claim your Bahamut esper.

It takes a while, but he'll die. If you need to figure out how to find him, get in the airship and angle yourself to the right or left VERY slightly and then go forward. On the map, your blip should appear to be scanning the map. You'll run into him eventually.