It seems pretty clear to me that you wield a Claw in 1 hand. But I don't think you wield a Bite or a tentacle in 1 hand.
I don't think even a Claw is either a 1 handed or a light weapon. For instance, you normally couldn't use the 2 weapon fighting style to fight with long sword and claw, treating the claw as your light, off-hand weapon.
Also, the OP's feat doesn't only work on light and 1 handed weapons, but rather "with any slashing melee weapon you wield in one hand," which is not the same thing as a 1 handed weapon.
Funny you should say that you can't use a claw as an offhand attack with two weapon fighting.
Natural Attacks: Attacks made with natural weapons, such as claws and bites, are melee attacks that can be made against any creature within your reach (usually 5 feet). These attacks are made using your full attack bonus and deal an amount of damage that depends on their type (plus your Strength modifier, as normal). You do not receive additional natural attacks for a high base attack bonus. Instead, you receive additional attack rolls for multiple limb and body parts capable of making the attack (as noted by the race or ability that grants the attacks). If you possess only one natural attack (such as a bite—two claw attacks do not qualify), you add 1–1/2 times your Strength bonus on damage rolls made with that attack.
Some natural attacks are denoted as secondary natural attacks, such as tails and wings. Attacks with secondary natural attacks are made using your base attack bonus minus 5. These attacks deal an amount of damage depending on their type, but you only add half your Strength modifier on damage rolls.
You can make attacks with natural weapons in combination with attacks made with a melee weapon and unarmed strikes, so long as a different limb is used for each attack. For example, you cannot make a claw attack and also use that hand to make attacks with a longsword. When you make additional attacks in this way, all of your natural attacks are treated as secondary natural attacks, using your base attack bonus minus 5 and adding only 1/2 of your Strength modifier on damage rolls. Feats such as Two-Weapon Fighting and Multiattack can reduce these penalties.
Note how it explicitly states that you can reduce that penalty for natural weapon attacks made in conjunction with manufactured weapon attacks using the two weapon fighting feat.