@Riger He was going to let it go, and in fact had already given up and was preparing to leave, but plot kept him in the village a little bit longer, and ultimately she joined.
So the answer seems to be yes. If she just tells him flat-no, he will eventually call it a loss and move on.
Also, saying "my sword is better because of experience" feels like an excuse. His sword is better because he was taught bullshit alchemy-construction magic by a bullshit hyper-wizard. If she could use magic to rearrange the fundamental properties of matter and energy, I'm sure her sword would be significantly better, too.