I'm only answering this because you are specifying a lack of funds for modern medical approaches. Don't feel helpless. Lots of kids have ADHD and lots of kids will act silly as a form of misbehaving, even if they don't have ADHD. My gym has a karate class I've been scoping for my son, and I see boys act this way often. Same at the soccer field, in school, etc.
In families where this is not allowed in no uncertain terms (or in some rare cases where kids are born easy), a warning will suffice. This does not cause anxiety because the behavior does not escalate to the point where the child is being embarrassed and corrected by teachers. Even kids with ADHD will respond to calm effective discipline. Often they are MORE relieved by the clear, structured boundaries. My friend's son with SEVERE ADHD behaves much better at his dad's who runs a tight ship than at his mom's. Your son feels embarrassed when corrected by instructors, and he is able to behave until he decides not to, so he really can control this and he does understand. If you don't feel that discipline will cause anxiety, it will work for you, but if you feel that it will, then you won't use it.
You know his past behavior and you know the signs, the face, the posture, the attitude. When he is in class, and he very first begins the FIRST few seconds of attempting sillyness, you should give him ONE calm warning to stop. If he continues for one more second (before he manages to roll around making noise or flop on the floor etc), you calmly walk over to him (the teacher will not mind-warn the teacher up front, he'll be relieved) take his hand and remove him to the restroom for discipline with clear explanation of how he is disrupting the class. If he writhes and screams that you are removing him, restrain and calmly, with a peaceful face, carry him out. Explain, and Enforce Firmly. Let him calm down, end it on a positive note. Tell him everyone likes him, he's a great kid, he's doing great in the class, and let him go back to the class and act well. Repeat several times if necessary, but don't let the sillyness proceed long enough for the teacher to have to address it. If you do this every time, he will NOT continue this. If he cannot respond to a situation this clear, he may be unable medically to participate in the class, but it sounds to me like he can do very well in the class.
In my daughter's swim class last year, she was new to swimming, felt afraid, didn't know the kids, the teacher, etc. She wasn't acting like herself. Before I knew it, I heard the teacher repeatedly calling my daughter's name (she was then 4). When I finally located her in the pool full of kids, she was floating away from her group and playing on her own and not doing what the teacher said. ??!??!?!? I warned her, "Sweetie, listen to the teacher right now." SHE DIDN'T!!! SHE STARTED CRYING! WHAT? I started looking for ways to walk around the pool to where they were with my other two toddlers in tow so I could handle her, but somehow the teacher got her under wraps quickly and the lesson went OK so I didn't do anything. When we left I EXPLAINED what was absolutely not allowed, but I KNOW that the explaining does nothing for most kids this age, even good ones, and ones with no ADHD. So the next day, I warned her right before we went in how to act. Again, even in well behaved kids, this is usually not enough to insure a bad new habit has been nipped, especially if they have gotten away with it once. So. I asked another mom there if she would watch my other two for a second "if necessary". Sure enough, 10 minutes into the class, my daughter was floating away from the group and making a weird pouty face when the teacher called her. I handed my younger two off and bolted to the side of the pool like lightning. I said, "Sweetie, come here please right now." I removed her from the pool and brought her to the restroom. Calm as the afternoon breeze I specified that she was wandering from the group and not listening to the teacher again. She apologized and said she would be nice in the class. BECAUSE I had been consistent with discipline her whole life and because she was usually extremely good, and because I could see she "understood me" I didn't need to enforce in that moment, but she knew I would the next time. She had no further wrong behavior and was great in the class after that. But I know for a fact had I not addressed it forcibly she would have continued like LOTS OF THE OTHER Kids in the pool!
On a different note, ANOTHER little boy cried the whole class long for the whole two weeks. His mom sat at the picnic tables with us explaining how he had anxiety about water since birth and she has no idea why he acts this way at 4, they have a pool and he's had lots of lessons..but anxiety this and that. Coincidentally, to an outsider, as a side note, she ALSO didn't HAPPEN to be doing anything about it. Same thing in the locker room, he was crying because he didnt' want to leave, and screaming that his brother took his sandal, and no response from mom other than comforting. So who really knows if it "couldn't be" stopped. But interestingly, one of the days, his dad brought him to class. And would you believe? Not a peep. The water was his best friend.
You sound like you already handle this very well. I think if you step it up you'll see results. You're doing the right thing by having him in the group activities and it's only natural it's harder for him with anxiety and ADHD which puts more on your plate, but you can help him through this. Enforce his right behavior, and he will learn it. Your only other choice is to let it continue and watch him be ostracized like you are fearing. Good luck, you can do it!