Does it say "Melee Basic Attacks also push 1 on a successful hit" or does it say "When you hit with a melee basic attack, push the target one square."
In the first, it's an augment, in the second, it's a trigger.
If it's a trigger, then it's Lashing Flail providing the push effect, not Melee Basic Attack.
If Lashing Flail is providing the push effect, then it's not a power delivering the push.