Anything could happen, but I wouldn't expect problems leaving Australia; the inspection is usually reasonable. In the US, though, it's very much a matter of who's manning the checkpoint and their training/attitude/phase-of-the-moon...

You have to collect your checked baggage when you arrive in the US, go through customs and then re-check it, so you could take the water bottle in your carry-on for the international flight but then put it in a checked bag before the US domestic flights.

I would think that it would okay in checked baggage. That is, anything can go missing from checked baggage, but there's no particular reason that they would either single it out as a threat or want to steal it.