Part of the reason I prefer not to put the spent air stream into the flare header is oxygen, H2S and liquid water will form elemental sulfur at ambient temperatures. Flare headers nearly always have H2S, so the likelihood of forming solid elemental sulfur is quite high. Is the sulfur in the form of elemental sulfur? That would indicate H2S breakthrough from the caustic treating system, which should not happen unless the caustic gets completely spent. This seems unlikely as the mercaptan removal would suffer before the H2S broke through.
The spent air stream should not be as corrosive as you indicate. What is the operating temperature of the caustic treating system? What is the oxygen content of the spent air? Excessive oxygen can aggravate corrosion.
My preference is to still route the spent air stream to a fired heater or incinerator rather than to the flare.