You can get native-style scrolling on an HTML element with overflow by using the following proprietary CSS property:
-webkit-overflow-scrolling: touch;
It has some caveats, though. Depending on what’s inside the element, rendering might be slightly broken, so you should test it and see if it suits your particular needs. I’m also not sure if it works properly when you specify overflow-y: hidden. If it doesn’t, you should be able to get it to work by playing around with different values for overflow-x, overflow-y and overflow (auto doesn’t seem to work).
If you need to, you can fake overflow-y: hidden on your div by creating a second nested div with the content and setting that property on it. But I hope that’s not necessary.