In Kao Round 2, I can run forward with my analog stick just fine, but if I try to turn the movement slows down to walking speed. I think it's because my analog stick gives a circle of x/y values and the game expects a square. Is there a way to fix this so I can run in the game properly? I am using an 8BitDo Pro 2.

I am not aware that Kao offers an option to select the gamepad type, I just use a old gamepad (Logitech Rumblepack 2) with square movement for such old games from that era (others being Lego Star Wars 1 and Ultimate Spider-man for example, they have the same problems with modern gamepads).

However, there are some gamepads like the EasySMX ESM-9110, that have an extended movement cirlcle to solve that problem. The diagonal extension is enough for the games to set the movement to full speed.
This gamepad is currently my favorite for all kinds of games, also has autofire and some programmable buttons on the bottom.