Hey, K....
I know it's more expensive, but good shoes for flat-footed kids are essential. I have flat feet, too, and I've spent $100 on good shoes. Granted, that was last year, but maybe you could try the arch supports for now, which fit into any shoes (as another poster suggested). Just keep in mind, though, that they may not necessarily fit comfortably into every shoe. So, still have your son try shoes on WITH the arch support. (When I was a kid, my arch supports were from my doctor.)
You could also try Kid-to-Kid consignment for good shoes or Ross (they even have Sketchers for like $12-20 for kids.) So, they are affordable, stylish, and comfortable. You may want to also talk to the store about their return policy with their shoes and explain the situation. I have certain brands that I love and look for when shopping for shoes, because I know they will be comfortable and have good support. So, maybe you could ask your dr. for some recommendations for brands.