I'm sorry to hear you're going through this.
If you were wanting to do a sleep study, my husband did one through Genesis Sleep Care. They work out of several locations including Kawana on the sunny coast - both of his sleep studies were done in Milton/Auchenflower. We have private health so the only fee we had to pay was the hospital excess fee.
It might be a good idea to go see your GP - they can refer you to a sleep clinic if needed, and they'd also be able to refer you to one that would bulk bill if they exist. If you need a recommendation for a brilliant GP and are happy to travel in to the city, PM me as mine in Spring Hill is wonderful! The only disadvantage is that the clinic doesn't generally bulk bill.
DH does have sleep apnoea, and since having the sleep study done he's got a CPAP machine. He doesn't snore like a freight train anymore, and he's been getting some fantastic nights sleep!