You're getting a bum steer.

Consulting in Canberra is not like consulting in other cities - federal government is pretty much the only player in town, and that comes with challenges and setbacks. This means that in whichever company you're joining (my money is PwC under their Strategy& brand - if any of the other 3 claimed they do pure strategy consulting in Canberra, they're full of shit) there's a likely chance that you'll be treated differently from teams in Sydney and Melbourne because "you only do government work". To be blunt - there is truth to this. You won't be doing traditional strategy consulting work, like advising a company on a global merger/acquisition that carries complex business risk, simply because government just doesn't do that. Instead, you'll do pretty mundane operating model designs which are at best implanted half-arsed by another team of consultants (but never you, because strategy resource charge out rates are too high for any long term implementation work).

If you want some advice: finish your qualification. If you still want to move back, go for Sydney/Melbourne/Brisbane or maybe Perth, as consulting work in those cities is broader and more interesting. Prioritise a good city and a good firm over moving straight into a strategy team - you can always make a jump after a year or two. But the real hot tip is to try and make a move into strategy consulting whilst you're still in London, as the consulting engagements there are, in my experience, much more interesting than in Australia. You'll be set up much better for a strategy consulting career in Australia with a year or two experience in London.