It seems to me a Klingon cook would be at best unwelcome on the station once the Cardassians took it back (however briefly).
Was it ever established if he left with the original evacuation or stuck it out?
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He never shows up nor is he ever mentioned again after Season 2. MemoryAlpha has some trivia on him though:
The Klingon chef was played by the late actor Ron Taylor. In several novels, the chef is known as Kaga (β). The name may or may not have been applied as a tip of the hat to Chairman Takeshi Kaga, host of the Japanese cooking television series Iron Chef, which debuted the same month that "Melora" initially aired.
And this from MemoryBeta:
In 2372, Worf dined at Kaga's restaurant, where he enjoyed live gagh and a rokeg blood pie for dessert. (DS9 short story: "Broken Oaths")
I can't find anything else about him. It seems it was just a bit of world building to establish DS9 as a melting pot of Alpha/Beta Quadrant cultures.
If it exists, you can buy it at Deep Space Nine. Even live gagh.