Ive asked a psychiatrist to bulk bill me before. It worked but the therapeutic relationship was challenged and I changed doctors.
I am bulk billed by my family GP and so no out of pocket costs, he does my scripts, however the prescriptions I have for depression/anxiety are minimal at best so he is happy to prescribe them. There are many prescriptions GPs aren't willing to touch, they would prefer to involve a specialist such as a psychiatrist.
Do you have any community centres that provide no/low fee psychiatrist sessions? Perhaps your current psychiatrist could refer you so as to avoid huge payments?