I’m with ANZ for my mortgage and a credit card came with it. We pay it off every month no matter what. We use it for everything. It’s a frequent flyer (qantas) card so we’ve racked up hundreds of thousands of points, which is now getting our international flights later this year.
If you trust yourself to pay it off consistently, then they can work in your favour.
This seems like a strange question, not sure I’m understanding. Usually people hesitate to open extra credit lines before getting a mortgage, but if you already have that then what is the concern? Do you not already have any credit cards?
You probably did not need to cancel those cards, and I’m surprised to hear you worry about dishonesty. What would be dishonest? You are allowed to have debt or credit with more than one company.
As long as you are honest on your income and debt declarations, go ahead and apply for the credit. If they don’t feel you could repay it, they won’t extend the credit. Those companies know everything :p
It's unlikely to matter much but depending on your monthly spend you might see some tiny gains over the life of the loan if the account you pay it off with is setup as an offset on the mortgage.