They may be able to do it under a twilight sedation or a GA, and it's not like being "cut into" at all, because your mouth is open. I had a wisdom surgically removed in the oral surgeon's chair, it was pretty much like a normal visit to the dentists but a longer time in the chair, but with more of a recovery period (and more painkillers) afterwards.

FWIW I recently had some very nasty oral pain (nerve dying inside a tooth) and the specialist said the most effective painkiller combination is to take nurofen and panadol together. You might get more mileage out it than the panadeine forte.

BTW if the specialist doesn't call, you call them. Sometimes the urgency gets lost in translation, so if you call to follow up you might get in sooner.