Makes me wonder if Shirou could trace Fragarach.

He cant, at most he can make a hollow copy without any power. Fragarach is a divine weapon.

Shirou has Archer's combat experience, so whilst Bazett has higher stats with Rune Amp, Shirou would have more experience, and has beaten far stronger enemies.

Bazett easily has the same, if not more experience, she has been doing really dangerous enforcer jobs since 15 and the targets of the enforcers are always OP not for anything a lot of times Bazett and Kirei teams were wiped out.

Why Kuro went for Caldbolg (which Bazett literally caught, that was a WTF moment) and not Hrunting, i don't know. Shirou isn't too weak to use NLBW properly like Kuro was, so that would probably be enough.