If you go private for the first consultation then you only need the name of the consultant who runs the clinic that you want a place on. A lot of (all?) the other specialists you could see will report to him / her.
So, s/he can order the booking staff to find your DH a place in the clinic, if he needs it.
At least that's what I've experienced when I needed a referral to a specialist and was lucky enough to have private insurance at first. All you need to do is call the administrator who told you there are no spaces until September and ask the lead consultant's name. Then ring the hospital and ask for that consultant's secretary and ask her about availability (and cost) for a private consultation.
It is not queue jumping (if that's what you are thinking). Its getting your DH correctly triaged. He won't get an earlier place in the queue if he doesn't need it.
It's a shame that sometimes you need persistence and sometimes a bit of money just to get what you need from a supposedly free at the point of delivery service, but that's how it is,