Not gonna vote (since ofc I'm the OP, lol), but Kirito could figure out a Shadow's weakness imo. He has a pretty high battle IQ. He already has a superior base attack potency (Since a 3 floor difference in SAO is already considered a stomp match, or around one shot territory), and Sword Skills boost that even further (up to one shot territory, as Sword Skills can normally one shot an enemy that would take multiple blows otherwise).
The Search Skill notices minute details anywhere in the user's field of vision. He'd be able to see a moving shadow, especially since the Floor 5 boss room is an open arena, and especially if they start only 5 meters away, where he'd be able to clearly see her drop into her shadow.
Also keep in-mind that VR avatars aren't the same as normal humans, they only take the shape of humans, kinda like Shadows actually. Strangulation has never been shown to be possible in the game, and everything is based on HP, as their HP goes down when they're underwater. They also have no organs or insides, and slashing through him or trying to crush any body part won't work either because, well...
So this means traditional means of killing a human wouldn't kill Kirito either, she'll have to widdle down his HP, and he'll of course always have potions to heal back up.
As for how Kirito could figure out a Shadow's weakness, he'd successfully hit her with a Sword Skill, as Sword Skills also massive boost one's speed, and most gap-closing Sword Skills are exactly 5 meters. When she immediately regenerates, he'll immediately be thinking Shadow Mio is some type of mob with a gimmick, as there's tons of gimmicky mobs on SAO, such as the Irrational Cube, which requires solving a Rubik's Cube + multiple Sudoku Puzzles in a row to defeat, and thus Kirito would immediately be trying to look for weaknesses.
The moment Kirito sees Shadow Mio dip into her shadow, or disappear and damage him from a blind spot from within a shadow (which he'd survive via his VR physiology, his HP would just dip), he would put 2 and 2 together that her true self must be the shadow, rather than the person themselves.
The only thing that Kirito would struggle with is Shadow Mio's maneuverability advantage, especially since she can scale the boss room's walls, though I don't remember any time in the series when a Shadow scanned someone while inside of their shadow. Even Haine needed to be manifested as Shiori to copy Shinpei.
Also, I'm not too sure if Shadow Mio would be able to copy the game mechanics of SAO, and thus be able to use Sword Skills. A Shadow is just a projection of a person after-all, not the real deal, so she would still have her stats instead of Kirito's, though having Kirito's memories may be beneficial, but at that point, he'd likely have her weakness pinned down and strike it via a feint or other means, since Kirito's whole style is based on tricking people and creating openings once he finds a weakness.
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