Is it possible for a creature like a bear to make an attack that counts as an unarmed attack, even though it has a claw attack as a Natural Weapon?
An Unarmed Strike is (kind of)1 a natural weapon.
Not all natural weapons are Unarmed Strikes.
Natural weapon is a category of weapon, which includes Bites, Claws, Slams, Stings, Tail Slaps, and yes, Unarmed Strikes (sort of).1 Improved Unarmed Strike improves only Unarmed Strikes, not the rest. Improved Natural Attack can improve any one natural weapon, except not Unarmed Strikes because Paizo hates Monks even more than Wizards did.
Note that Feral Combat Training changes this somewhat, however. Improved Unarmed Strike itself still doesn’t apply to natural weapon other than Unarmed Strikes, but anything that has Improved Unarmed Strike as a prerequisite does apply to the chosen natural weapons when you have Feral Combat Training, even if usually they only apply to Unarmed Strikes.
Ultimately, you’re burning two feats to absolutely no effect other than possibly making other feats work. If you found some really good feats that improve Unarmed Strikes, it might be worth it, but I doubt it. That’s a ton of feats. And honestly, I think a good DM should just let most Unarmed Strike-improving feats work for other natural weapons; this is not exactly a high-power strategy here. A two-feat tax is wholly unnecessary.
1 I say “kind of” because Unarmed Strikes are really weird. They mostly follow the rules for manufactured weapons, not natural weapons (e.g. you get iteratives, you don’t get multiple Unarmed Strikes, etc.), but most spells and feats treat Unarmed Strikes as natural weapons. And on top of that, Paizo put a further restriction on Improved Natural Attack to prevent it from working with Unarmed Strikes. So Unarmed Strikes are very much a weird hybrid case. In any event, Claw attacks are not Unarmed Strike attacks.
The damage from an unarmed strike is considered weapon damage for the purposes of effects that give you a bonus on weapon damage rolls.
An unarmed strike is always considered a light weapon.
Benefit: You are considered to be armed even when unarmed —that is, you do not provoke attacks or opportunity from armed opponents when you attack them while unarmed. However, you still get an attack of opportunity against any opponent who makes an unarmed attack on you.
A monk’s unarmed strike is treated both as a manufactured weapon and a natural weapon for the purpose of spells and effects that enhance or improve either manufactured weapons or natural weapons.
In regards to anyone other than a monk, an unarmed strike is a light weapon that is neither natural nor manufactured.
Exclusively for the monk (and others with monk-like abilities), an unarmed strike is a manufactured light weapon, and a natural light weapon.
What is a Manufactured Weapon?
Any weapon that is not intrinsic to the creature, such as a sword, a rock you pick up, a club you make.
What is a Natural Weapon?
Natural weapons are weapons that are physically a part of a creature, such as a bite, a claw, a tail slap.
Unless specifically mentioned, (such as the Monk, creature listing or racial description, etc.) no one has a natural weapon right away. For example, no human has a claw attack, a bite attack, etc. without feats, spells, or other abilities that gives that to them.
Also, an unarmed strike does not cause lethal damage, provokes an attack of opportunity, and is a very unappealing option without Improved Unarmed Strike, and/or some other way to make it better. Monk's specifically mention that their unarmed strikes count as both manufactured and natural or intents and purposes - because almost no one else's do.
If you take Skip William's advice to heart (which is not RAW) then you will find him stating, "For purposes of weapon enhancements, an unarmed strike is considered a natural weapon," and "Natural weaponry deals lethal damage without recourse to a class feature or feat, such as Improved Unarmed Strike."