For some reason I thought the OP was going to Sydney

I've traveled with child seats before. I put them on the plane seat with the plane seat belt to hold it down and then used them in taxis, hire cars etc. I've also seen people putting them through with their luggage.

I've had some very dodgy seats with hire cars over the years. Some that looked like antiques and some that have been totally age inappropriate (a booster seat for a 6 month old ) and one that smelt very strange.