In your case, I would suggest "Yes, put Javeer Baker, Ph.D" on your card. You could optionally put "in social science" on it if you'd like.
I cannot speak about the culture in Australia, but in the United States and Canada, it is appropriate to put your title (Ph.D) on your business card if the industry generally is supportive of or respects academics, or in which high degrees are not common. I might guess Australia's culture is similar.
Industries that would want to be proud to have a Ph.D on staff especially would be non-profit organizations, advocacy groups, research think-tanks, and consulting. As you said that your employer was an advocacy group, I believe that putting your degree on your business card is beneficial.
Industries in which you would NOT put your degree on your business card would probably be software high-tech (because no one cares) or in academia (not because no one cares, but basically any assistant professor and up basically has a Ph.D and therefore it's not impressive), or, as mentioned above, a construction management company or similar.
(As an aside, a whole bunch of people at a conference I was at rolled their eyes when they saw one of the authors wrote "name, Ph.D" on a paper.)