Ultimately, the errata clarifies Snap Kick so we don’t have to worry about this any more. With the errata, the feat’s benefit reads:
Whenever you initiate a strike or use the attack or full attack action, you may take a −2 penalty to attacks made this round gain an additional attack at your highest attack bonus (the −2 applies to this attack as well). This attack is an off-hand unarmed strike, and deals damage as appropriate.
Instead of defining the damage in the feat, it just says it deals off-hand unarmed strike damage. “Off-hand” here is actually a little awkward (because you only really “have” an off-hand while using two-weapon fighting, which you aren’t here), but it’s clear enough what it means—halve your Strength bonus to the damage roll. This result winds up being identical to what Snap Kick originally said explicitly, but now it’s much clearer that this figure can be changed by anything that changes one’s unarmed strike damage and that this should be treated in all ways as a regular unarmed strike attack, albeit one you wouldn’t otherwise have gotten a chance to make.
However, even without the errata, the attack made by Snap Kick was defined as an unarmed attack, and the damage calculation offered by it matched that used by an unarmed strike made as an off-hand attack while two-weapon fighting. This was interpreted by most as simply a reminder of how unarmed strike damage works, rather than a fixed figure that cannot be changed. If it were fixed, then like insightful strike, it would explicitly say so. It’s also unclear what it means to be “an unarmed attack” if your bonuses and improvements to unarmed attacks don’t apply. The errata, effectively, confirms that this reasoning was correct.
So yeah, Shadow Blade affects your extra attack from Snap Kick, adding your Dexterity bonus to the damage roll.