I had a Versa rental for a trip through Northern California a couple years ago, and while I didn't use the same route you're talking about, I still did plenty of mountain driving. It does ok. You're not going to fool anyone into thinking you're a high powered sports car, but you're not going to actually struggle to move around either. I actually thought the transmission (CVT instead of an actual gearbox) was more problematic than the actual engine.
However, also keep in mind, you're certainly not guaranteed to get a Versa. You're renting a Versa "or similar," and that means you could get anything that your rental company considers to be a compact car.