The most economical would seem to be to change from LHR-SYD-AUH to LHR-AUH to be honest ( based on saying that you are travelling on BA to Sydney via Singapore to then fly on Etihad )

If you have an Amex Platinum Card, then there is an AMEX lounge in the international terminal , however , unless the EY flight is on the same itinerary as the BA one, BA will not check your luggage through

If you wish to try using the domestic lounge by purchasing a ticket , then you will also need to put luggage into storage

There is a security check at Terminal 3