So, yes, a tempest fighter does get to apply his bonus because the feat gives him the ability to wield non-off-hand weapons -as tho they were off-hand.- This is -exactly the same- as how a wizard can apply Wand of Accuracy to attacks made through a sword implement if he takes Wizard of the Spiral. No, it's not a wand, but it is treated as though it were a wand.
While I'll happily admit that my opinion might be wrong, I don't think the situation is "exactly" like the wizard situation. The wizard treats his longsword as though it were a wand, yes, but "wand" is not a weapon property; it's an implement. So Wizards treating their swords as if they were wands would be the same as the multi-class ranger treating his sword as if it were a dagger (or handaxe or whatever) and taking all the hits to damage dice and the like. But that's not what the feat says.
The chainfighter feat, for example, gives the spiked chain the light blade and the off-hand properties, but the ranger feat is not worded in this way. It says that you can wield one-handed weapons in your off hand as if they were off-hand weapons, but doesn't say that one-handed weapons wielded in your off hand are treated as if they had the one-hand property.
Sure this is nit-picky, but I believe it's what's intended. In the same way, I don't think you get to treat a longsword or bastard sword wielded in two hands as a two-handed weapon for the purposes of powers and feats that reward bonuses for 2-handed weapons (like Reaping Strike or the Bloodclaw enchant).
It doesn't matter how you're allowed to wield them (in my opinion), if it doesn't have the weapon property (off-hand/2-hand) you don't get the bonus.
Again, however, this is only my own understanding of the rules.