Using 'the' in front of proper nouns?
Because there are two different Sydney domestic airports (as noted below) there could be confusion if you referred to "the Sydney Domestic Airport", because uppercase letters often are pronounced the same way as lowercase ones. There is also potential for confusion among persons who think of Kingsford Smith as an international airport, rather than domestic, and might think Bankstown Airport is being referred to in sentence "I transferred flights at Sydney Domestic Airport". All of "I transferred flights at Sydney Airport", "I transferred flights at SYD", and "I transferred flights at Kingsford Smith" are clear, relatively unambiguous, and grammatical.
Note, "Sydney Domestic Airport" apparently is Terminal 2 or 3 at Sydney Airport, that is, at Kingsford Smith Airport, coded as "SYD" and known as "Sydney International Airport". "Sydney Domestic Airport" apparently does not refer to the Sydney domestic airport known as "Sydney Bankstown Airport", coded as "BWU".