You mention he has a bowel movement once a day, but I'm wondering about the consistency of it. (TMI I know, but it's important) Is it soft and easy for him to go or is it like stones? Stone-like, or hard at all, is indicative of constipation.
Constipation can contribute to UTIs, but even with constipation that number of UTIs is a huge red flag for a child who needs further urologic testing. My daughter has had testing including VCUGs several times and never had any sedation for any of it.
Constipation issues along with urologic issues can also be indicative of something neurologic going on. Your comments recently about leakage could also indicate that or it could be leakage caused by pressure on his bladder from backed-up stool.
You need to get him further testing. Soon. I can't sugar coat that in any form. Others on this thread have said it, but if he does have reflux it can lead to serious health risks. Far beyond any concerns over circumcision (which you definitely shouldn't allow at all and it's an absurd suggestion) is the concern for kidney damage.
Here's a link
for a study on sedation during VCUG. You can discuss it with the ped uro. Many ped uros don't like to do sedation for VCUGs because of the concern that it will affect the results of the test. Also, since the test is relatively quick there are concerns about the risks of the sedation itself since it is not absolutely needed. You ped uro most likely will never agree to full anesthesia, but maybe you two can agree on some sort of light sedation.