If you can afford it then I would bring Charlie, the stress of re housing in UK would be the same as the stress of coming to Oz so what really is the difference except she will be with the family she knows, I agree you have to be a bit more careful here with the wildlife but it depends where you will be living and if she likes to explore...
We left our dog in the UK, it wasn't a financial decision but more to do with the climate, we had a really hot summer in the UK the year before we left and she really suffered so we thought that Oz would be even worse... but we didn't realise most of the time is is a lot less humid here so she probably would have coped well, apart from the really hot days but then that's what air con is for.. we did regret it although she did settle very well with her new owners.
Its a very difficult decision and to be honest probably only you can make it as you know your cat.
Generally renting options with an animal isn't too bad either, further from the city tends to be better.
Good luck with what you decide