You did not specify this, but I have the impression that your son did not have these behavior problems with his teacher before, and it is only with the "replacement teacher" that he started acting up.
IF this is the case, then it is obviously the teacher's failure, and she needs to be trained to handle it.
Because this is a private school, however, they can pretty much do whatever they want! Parents are paying them to give their kids attention, and because your son is requiring so much...the other kids are not getting it, and so their parents might all start to complain! Well, then the school has a choice of disappointing/losing ONE parent, or a whole bunch of parents. Guess what they will choose?
In a public school there are certain rules they must conform to, and they can't kick kids out, except for repeated violence and other serious offenses.
You want reassurance that you are doing the right thing--in THEORY you are. I have a BA in Child development and I have worked in schools for several years (as a behavior specialist), and I have 3 boys at home. I know the theories. I also know that there are some exceptional kids who defy logic. Some kids are really unusual...as in amazingly stubborn and willful. Some kids have a curious mind and a strange sense of humor---they like to see how a person will react, and they will take it to the limit, just to find out.
All kids need clear boundaries...but some will cross the boundaries anyway, because they just have to find out what will happen, or because they know what will happen and they think it is fascinating and/or hilarious, and they don't care if they are punished.
One of my boys used to knock his big brothers' building blocks over a dozen times in a row, even if he got hit by his brothers (both scary and painful for him). He told me he really didn't like getting hit, and he knew he would be hit, but he liked knocking the blocks over MORE than he didn't like getting hit. That was his "cost/benefit analysis"--a giggle was worth getting hit.
As for what you can do (without knowing your child, and not knowing if the speech/social skill class is related to a serious social disability) all I can say for sure is that stickers aren't going to cut it.
If he cared about approval and stickers, he would have changed his behavior. If he doesn't defy you at home, he is clearly already CAPABLE of self control. Self control is not the issue---he is making a choice to act this way with this teacher. He is getting something out of it, and you may have to balance that with something negative.
If you are not sure what will specifically motivate him, or what he is trying to gain with his behavior, you can just be very strict---meaning zero tolerance. BIG consequences immediately and consistently, every time (like the whole world turns upside down for him).
When he sees that life is bad for him on days that he acts up, he'll change, because he'll see that it (whatever enjoyment he gets from messing with the teacher) isn't worth the disaster that follows at home.
Whatever is a disaster for your son, only you would know---for some kids it is endless lectures with them as a captive audience, for others it is being left alone (no talking to them or looking at them, or even being in the same room), or for others it's having to do endless chores, or losing all their toys--or some combination of all of those.