Switching wheels from 15x5 to 15x6

The stud lengths wouldn't be need to be changed, it's the offset or backspacing of the wheels you need to be mindful of.

Also, how are you getting bent wheels, unless you are hitting potholes or something?

SM-G781V ?

 

Also, if you want the tires to be more responsive during cornering, you want lower profile sidewalls, that would have less flex during cornering than taller profile sidewalls.

For that you would need to get larger diameter wheels and tires like 17 inch wheels and tires with the same overall outside circumference as your oem tires so your speedometer and odometer are accurate.

SM-G781V ?

 

Slow down a bit, especially with speed bumps, sunken manholes and the like. And sleep on the tire change?

 

FWIW, I've never been able to bend a steel rim. Scuffed them up and popped tires running them into curbs - but never bent them to a point that they don't balance or hold air. I've even put a dent on the tire bead, once - pounded it out with a hammer, replaced the tire, and balanced.
Here's the data you requested.
2015 Toyota Prius c - Wheel & Tire Sizes, PCD, Offset and Rims specs | Wheel-Size.com

 

Oh, my wife found a way to bend a steel rim. I asked around about a wheel shop to get it trued later, but (at least in my area) that apparently went out of fashion in the 70s, everyone said "just buy a new wheel, they're cheap."

I guess I would not think of "wider tires" as the go-to move for "turns instability" issue in a Prius c.

...I'm not saying I know of a magic fix either.

If anything I'm curious about what exactly "turns instability" means.

I've driven many miles in Los Angeles, but never in our Prius c- and I'm not sure I'd want to. All those expansion joints on LA freeways are uncomfortable in medium-wheelbase cars, and probably even less fun in a short-wheelbase car like a c.