I taught autistic students for 30+ years, and I hesitate to give you advice. These kids are so individual that there is no 'one size fits all' solution.
We found that not only was there discomfort associated with actual defecation, but things like the toilet itself could be a problem. (Too high, too cold, too noisy -- the room 'echoed', at school -- florescent lighting -- the flickering was aggravating.) So you might try a commode, a wooden seat, low light, dampening the sound -- perhaps with music. We used a potty, but it was built for us by the school's special needs department. It angled back and had vinyl cushions -- seat and back rest that were washable. The potty was low and we covered a disability-sized washroom stall with blankets and a rug, turned out the washroom lights and used a flashlight (there were no electrical outlets -- a dimmer might work for you) for more ambient lighting. We brought in music, played at the volume our student prefered. We ran water as well.
These are general ideas. Honestly, advice over the internet is nearly impossible. I found that what worked this time, often need 'tweaking' for the next time.
On edit: If your child go to a school with a special needs or resource teacher, that person might be your best bet for hands on experience. Rewards and praise are fine, and anything the doctor has suggested -- but this is uncomfortable for everyone. I'd get on a wait list and tell the receptionist that you'd take any appointment/ cancellations for even one session for advice. I well understand your frustration and fear that your child is not getting the help he needs.