Conceptually, the bonus attack is a haft strike, and is thus more akin to fighting with an off-hand weapon than with a great weapon. Thus, it makes more sense that:
- The bonus attack does not have reach.
- The bonus attack does not add STR modifier to damage, unless you have the appropriate fighting style.
- The bonus attack is not made with a two-handed weapon, so neither the great weapon feat nor fighting style would apply.
At least, that's an argument which makes sense to me. Considering the alternative - that this combination would allow you to get the best of both worlds, by applying all of the benefits of great-weapons to the primary benefit of dual-wielding - this seems like the preferable interpretation. It's no fun for anyone if combination of weapons and feats is so far-and-away better than any other alternative. (Personally, I care more that the mechanics are consistent for off-hand strikes.)
Note that 3.5 spelled this out explicitly, and using a double-weapon as a two-handed weapon (for the purposes of Power Attack, and extra Strength damage) required you to forgo the ability to attack with the other end of the weapon.