Whether ULCCs go for E3s or not is largely irrelevant for anyone still in Australia. You're pretty unlikely to make that happen without meeting anyone face to face. One ULCC did sign off one, and only one, who was in the right place at the right time. Regionals are still the key to getting into the industry over here.
If you're hell bent on shooting for a non regional, which I don't understand why you would be, Breeze would be the best target IMO