Don't know if these are pet friendly or not but I do see a lot of them on the road.......presuming this is the sort of camper to which you are referring. There is an office in Brisbane which should suit you Jenr.
I have just joined. I am in a similar situation to yourself. I want to know what it is like travelling alone and my dog is also called Lucy. A kelpie. I am trying out a 10day rental. Try www.salamandatravel.com.au they accept pets in some of their vans. You can just type in different dates to see what the price is. Price will vary depending on when you travel. Winter times seem to be cheaper which is fine because we don't want it too hot for our furkids. I think they are actually based in Queensland. But they have depots Australia wide. Cheers Jacqui