No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
is the late a mortgage late...
My guess is that the file as a whole is lacking and this is the easy excuse they are using. 1 non-mortgage late will not stop a loan usually unless they are looking for a reason to say no.
AU will kick out a mortgage late within the last 12 months
this is going to be very difficult
good luck - I would call around to all of the banks and just ask