Yes, I think indicating Alice is an experienced detective is a good idea. Is there a tension point that having experience brings out? Is she jaded? Maybe that as she trained in the UK she has a different way of doing things and ruffles the feathers of the local constabulary. If you include this description in your intro letter you should indicate why.
That’s good you want to return to Australia! I live in Adelaide. We have been mostly sheltered from COVID but I think with our state borders opening up we should get a surge in numbers in the next few months. Likewise I want to return to the UK! Last time for me was ‘96 (sniff …). I lived in Wales for a while in the mid-80s.
PS: Pls disregard my ramblings if it all sounds like balderdash. You know your work best!