Pediatrician isn't going to be easy to convince to test. Using safe natural approach is the way to go and treat based on symptoms (giggling, stretching and trying to stand on head, jumping, etc.). Best to do stool test but the indirect test using organic acid test can be useful as well. Some of this specialized tests can be ordered without doctor requisition from sites such as mymedlab.com, etc. caprylic acid in MCT oil or coconut oil is very effective for controlling yeast at least for us. Good luck.

First off, I would assume that your pediatrician probably doesn't know about candida and autism... and you may want to search for a doc who does integrative work (would google MAPS doc autism, or integrative doc autism, plus your location). Our doc ordered a stool culture from a good lab (Great Plains), as many labs don't know how to test for it. In the interim, I would consider doing some natural anti-yeast intervention for your son. I would consider going gluten free/casein free/plus low carbohydrates, and add in probiotics if you haven't done so yet. One great way to fight candida naturally is to buy some oregano essential oil, dilute it at least 20/80 with olive oil, and rub a few drops on your son's feet twice a day.
Good luck! Hopefully, others will kick in with what worked for them.

We have done prescription (nystatin and diflucan) as well but found diet and natural anti fungal is the best way to manage. It is impossible to get rid of yeast permanently as it is likely related to immune dysfunction (eg low neutrophils) but it can be controlled.