Talk to your wife's surgeon as he/she co-ordinates the treating team. Say that you would prefer to go public for rads and ask for referral to someone in that system - if your surgeon isn't keen, ask why. (I didn't know to ask about public/private but my surgeon was very aware of my financial circumstances and brought it up with me.) Your wife will have a long term relationship with her surgeon and oncologist. Unless tnbc is very different to other bc, she is likely to see her radiation oncologist 2 or 3 times in total...ever.
I'm not familiar with Brisbane's facilities - others may be able to comment - but I haven't heard that private is any better with regard to equipment or treatment. And from what I'm aware, there's just as much appointment shuffling and waiting in both systems but I guess that could be different in different places.
I think you said that you have a friend in a local radiotherapy dept - if you feel you can without prejudice, then ask that person what they think.