Visa Application Fees for Rewards Points?

I recently applied for a visa and made payments to department of home affairs where I actually incurred quite a decent amount of surcharge.

I just found out that my credit card company doesn't count government payment to be eligible towards rewards points. However, when I called the bank up twice today, the two different ladies on the phone said different things.

One said that any transaction other than ATO payments and gambling payments should count towards points, while another said that any payment to government agencies doesn't count towards points.

Does anyone have any experience with this?