Hi, Daniel! I'm afraid of loud noises in public restrooms too, but I think we're talking about different things.
I think it'll be hard to gauge the noise factor of the toilets at any location at a glance, so I'd recommend avoiding the main public restrooms if you're able to seek out some alternatives. My top two suggestions would be the companion/family bathrooms and the baby care centers.
Companion bathrooms can be found at First Aid locations and
throughout the parks. There's a chance that the toilet flush might be louder than you'd like, but at least it'd be predictable and minimal (rather than the ongoing chorus of other toilets at public restroom locations.)
Each park also has a baby care center which would provide a quieter environment as well.
When our oldest son (now 6) was 3 years old, he was anti a lot of loud things, especially fireworks. We countered this with earplugs. If worse comes to worse, you could pack some plugs in your pocket and bust them out if nature calls at an inopportune time - principally, if a public restroom is your only option.