My grandson is also on the higher end of autism, however he's not able to be in a regular classroom. Also what is similar is that because he's able to complete the work accurately in record time once he is able to focus we've wondered if he's bored. What his parents decided after months of trying to find a way to get him to do the work is to just tell him that he's "good with this, let's get it done. How fast can you do it?" After about a month of this, he now sits down at the table after he's prompted to do so and does the work.
My grandson is now 9 and in a special ed program in a school separate from a regular school. We've spent years trying to find ways to help him succeed. What it seems to me is that even tho he does need special assistance there comes a time when, just like everyone else, needs to know that often we have to do what we have to do. He needs to learn, just as we've learned that the quickest way to get out of the "pain" of doing something we just have to do it. Your explanation for doing it is right on. Doing it is practice and it also shows the teacher that you do know it.
As to the reading, I'd ask the teacher about it. Teachers give lots of positive feedback. It's good that he thinks he's advanced. Having success guarentees more success. His attitude should be reinforced. However, he is likely not advanced in a way that matters for classroom assignments.
It will help you know how to support him if you have a clear picture of where he stands and what he needs. Perhaps he needs to hear that he's advanced. But I suggest it won't be helpful to tell him that he's ahead of standards for being in kindergarten. He's good, yes but he's not better than everyone else. It's important to not compare him with other students. He's good all on his own. You can say, "'yes, you may be ahead because you play at home. You're a smart boy to figure this out." sort of response. "You're a good reader. I'm proud of you." You may already be doing this. And I feel like I'm not explaining this well.
I would ask the teacher if he's really advanced in reading. He needs to continue having encouragement and support with reading. His reading level may not be an indication that he's TAG qualified. It is good to have him tested tho. I'm not explaining this well. Sorry.