This is a faulty premise. Lyse isn't a hothead. She actually shows restraint more often than not. Hotheaded compared to a lot of other Scions who'd sit inside, reading and wondering just to triple check that they don't make a boo boo blunder? Sure. Compared to actual hotheads? Nah.

Her only real fault is being a bit brusque with her speech. I think the most "hothead" thing that she does in the entirety of the story is seeing Zenos and yelling, "You're a dead man!" Before attacking him, not knowing his strength.