I have the same problem with sweating and I'm also out on my bike a lot. A flat fabric belt bag works much better for me than those thick padded things - they always look too sporty anyway. I usually wear something in gray or black where you can't immediately tell it's practical. The most important thing is that the bag isn't too wide and doesn't hang too low, otherwise it really looks weird with normal clothes. Do you have any idea how big the thing should be - should it be more minimal or could it also be a bit more voluminous?
For materials I'd go with cotton blends or canvas, not those shiny synthetic fabrics. There are quite a few small labels that make urban everyday bags like that, they look way more high-quality than the typical sportswear brand belt bags. Also check whether the bag is padded inside or if your stuff just bounces around in it - sometimes that's more practical for your phone, sometimes it's annoying. And important: a sturdy closure so nothing gets lost while you're riding. Do you wear the bag in front or on the side?