The problem with that logic is that the bone augmentations do not boost your unarmed strike, nor do they make you stronger, they simply make your fists more painful due to the hardened bones in them. The second you use Knucks you already lose that damage bonus, and the same applies to elemental strikes. Since you do not actually hit them with your bone-augmented fists, they should not automatically be able to help you there so the RAI claim is rather arrogant to be honest.
However, since the ability projects your unarmed strike ahead, it's a fair argument that this means that even though the bone-augmented fists aren't what hits the enemy (but the projected strike), the entirety of the strike gets projected including the hardness of your fists.