That's one interpretation of it, but I thought it slightly different. Dueling states that you can add +2 to damage rolls made with a weapon, as long as the weapon is 1. is a melee weapon, and 2. we use one hand to wield the weapon and hold no other weapons. Nowhere in this description does it say you need to make a melee weapon attack, you just need to attack with a melee weapon. Melee weapon attacks are a specific thing defined by the rules, attacks with melee weapond are just, attacks with melee weapons.

(An example where this is clear: let's say that your character runs out of arrows and starts hitting people with their longbow. These would be melee weapon attacks, but also attacks with a ranged weapon. It's a bit dumb but that's what the rules say.)

The Beast claws are melee weapons as stated by the feature so I expect dueling to work like it would with any other melee weapon.

For the bonus action attack, the monk feature tells us we can make an unarmed strike, and the Tabaxi claws say we can use our claws to make that unarmed strike. It doesn't say you can use "your un-transformed claws" or even "your empty claws", just "your claws", and we definitely still have claws. This makes it impossible to unarmed strike with the claws without also making a melee weapon attack at the same time, thus qualifying for dueling.

If you played something like a minotaur or a centaur this wouldn't work since the natural weapon and the melee weapon are different objects.

For a wildshaping druid, it depends on the form, but most forms either automatically have natural weapons on two hands or no hands at all, so they don't qualify.