We haven't had luck recently. It used to be we could just send a message through our online banking account asking for a better rate and they would just knock 0.2% off the rate no questions asked. But then one day ANZ introduced online specials where if you refixed using their online thingy you'd get 0.2% off, and from that point on whenever we messaged they would just quote us the online refixing rate.

I gave up trying eventually.

With that said, there's no harm in working out what you can get yourself then asking a mortgage broker. We paid the carded rate for years for our flexiloan account (revolving credit) before talking to a mortgage broker. They couldn't get us better fixed rates, but they got us a deal on the flexiloan. I hadn't even considered trying to get the rate reduced. That was like a 0.5% drop or something.