We were in birdsville last year and there was no pet accommodation there. Maybe there are some towns that you will pass through that may have accommodation that are close to birdsville. Keep in mind the towns are many kilometres away. Try google. Otherwise call birdsville information centre or the caravan park there. I'm sure other travellers have had the same problem and they may be able to steer you in the right direction. Good luck! This is a great trip, you don't want to miss it!