The opiates are for post surgical pain, not the procedure itself. Post surgical pain can range from non existent to quite severe following wisdom tooth removal and the standard practice in the USA is to prescribe painkillers in case of pain, so oral surgeons don't have to deal with people calling them in severe pain later in the day I guess.
I've had all four of my wisdom teeth pulled as well as three other teeth and when I had my top two wisdom teeth pulled there was virtually no pain and I went to work the following day. When I had my bottom wisdom teeth removed, which were severely infected and impacted, the pain was so severe I don't know what I would have done without the codeine my surgeon prescribed. I had to take six tylenol 3s just to get the pain to be bearable and I needed a whole week off work to recover. Similar thing happened to my sister when she had hers removed, she was very ill afterward and in severe pain for several days. It is very variable how people respond.
If the OP is in good health there is a good chance he won't need painkillers but if they are needed the best are probably kava and kratom. Kava won't affect methadone PAWS at all, kratom probably will but a day or 2 taking a couple grams of kratom shouldn't be too significant in terms of setting back recovery. It's unlikely you will need more than a day or two of painkillers. I was only in severe pain the first 2 days after having my teeth removed. They continued to hurt for 10 days but it was bearable after the first 2.