I am also a mom with sun-friendly skin who has a very fair-skinned red-headed child.
My daughter (and my husband) flushes red as a result of the slightest exertion in the heat, which means I spend the summer with people asking me, "Is she okay?" And she is - it's just what she does. I make sure she takes breaks frequently in the shade and gets plenty of water. She still flushes deep red. So I've just accepted that this is her version of normal, and as long as there are no other signs of heat-related distress and she has been drinking water as she should, I don't worry.
Also, she is very sun-sensitive, so sunscreen is a must, but she also has skin that will turn red from irritation from the wrong sunscreen - based on our experience, high SPF mineral sunscreens or baby sunscreens are the least likely to cause a skin reaction which can be nearly indistinguishable from sunburn.
I have trained her to make peace with the idea that she will spend her outdoor time in hats. Fair skinned folk can sunburn their scalps right through their hair.
Have you changed laundry soap lately? Was the shirt freshly washed? Sometimes new soap plus sweat can cause a skin reaction for those with sensitive skin.
As for where to find a good hat - try REI or LL Bean. Or, if you can get your little guy excited about a cowboy hat, hit the IFA or a western shop. A real cowboy hat can be a bit pricey, but they're also really durable, which a hat needs to be if he's wearing it every day. Cabella's may very well have both western style hats and fishing hats to choose from.