Mostly just checking that I haven't misunderstood or missed something here (which happen frequently enough) but if I were to equip a Path of the Beast barbarian with two handaxes (or anything suitable for two-weapon fighting or Dual Wielder) and select the Path of the Beast claws I believe I can make four attacks a turn once I'm Raging (and have Extra Attack).

The Claws option for Path of the Beast states:

Each of your hands transforms into a claw, which you can use as a weapon if it’s empty. It deals 1d6 slashing damage on a hit. Once on each of your turns when you attack with a claw using the Attack action, you can make one additional claw attack as part of the same action.

It looks like I only need a single hand free to be able to attack twice with the claws, as the additional attack doesn't require it to be made using an off-hand or such, it also doesn't say that the additional attack must come after the Attack action.

Meanwhile two-weapon fighting states:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

So the only requirement is that at the end of the Attack action I must have a light weapon in each hand and attack with one of them so that I can then attack with the other as a bonus action (since this will need to happen after the action).

What I believe this should then mean is that if my barbarian has two handaxes, one held, and the other stowed in their belt or whatever, then on the second turn of combat (after initiating Rage in the first) I can do the following:

1. Make one Claw attack (one attack remaining in the Attack action).
2. Make the additional Claw attack (still have one attack remaining in the Attack action).
3. Draw and attack with the stowed handaxe (now holding two, Attack action is complete).
4. Make an off-hand attack with the other handaxe as a bonus action.

Hopefully I haven't misunderstood the rules around this as it seems to me like this should work?

The main drawback is that to continue this into a third round it seems like I'd technically need to drop one of the handaxes before the 3rd turn begins, so I can repeat the same sequence (picking the axe up at step 3 as a free object interaction), as a free hand is required for the claw attacks.

With some creative license smoothing it out I'm picturing a path of the beast barbarian burying two handaxes into his opponent, swiping twice with both "free" hands, then pulling the axes out (even though strictly speaking it's the opposite way around), or even throwing the axes (which two-weapon fighting also allows) before rushing in with the claws, but I believe both of these will be stretching the rules (or at least narratively reinterpreting the sequence of steps since the axes always need to be resolved last, and no throwing would actually occur).

Seems like it could be a fun option anyway, and not really exploitative as the classic greataxe + Great Weapon Master will probably eclipse it, though using Reckless Attacks for four attacks with advantage in one or more rounds would keep it somewhat competitive I think, even without any investment to get two-weapon fighting style and/or Dual Wielder?

That's an interesting combination of choices.  You could make it work better if your character was a Thrikreen, since they have four arms, so you could have your claw with a main arm, a shield with the other main arm, and hand-axes in each of the secondary arms.

Then for your Attack action, you could make your two attacks, one each with a claw and handaxe, which would fulfil the requirements for both of the additional attacks, one each with a claw and the other handaxe.  No dropping or picking up of weapons required.

This is something that has come up from time to time on the Barbarian class forum. But yes, once you reach fifth level you can use Two-weapon Fighting and get two PotB claw attacks.

For round 3 you just start with a hand axe attack. Then sheath the axe you attacked with to do your claw attacks. Finally finish off with the bonus action attack with the axe in your other hand. Now you are back in the state you were in at the start of round 2 and can loop as needed.

But really the advantage of PotB claws is they only need one hand to use. This allows you to do great damage while still having a hand free for something else, like a shield or to grapple an opponent. This makes ways of weaponizing your bonus action that keep your hands free especially valuable. Longtooth Shifter is the easiest but the Monk's Martial Arts and Flurry of Blows also work. Lizardfolk are also worth looking at for Hungry Jaws, improved unarmed strike, and natural armor.

Re read that Haravikk - for the axes you have to have one in each hand so no “empty” hands initially. However I think that by RAW you still could but the sequence would be:

1) attack with hand axe leaving it in target if you hit, dropping it if you miss for first of two attacks.
2) claw attack with now empty hand for second attack of the action
3) second claw attack with same hand as part of the claw attack.
4) bonus action attack with the “off hand” axe.
5) Free interaction to pull out/pick up first axe to be ready to repeat each round.

however, I suspect this is a glitch in the rules and as a DM I might well rule that when your hands transform into claws you can no longer hold the axes so you have to chose whether to use the claws or the axes and either way after L5 you would get 3 attacks (claw, claw +extra claw/ axe, axe + BA axe)

The transformation only happens if both your hands are free, it says;

Until the rage ends, you manifest a natural weapon.

It ->COUNTS AS A SIMPLE MELEE WEAPON<- for you

Claws:

->EACH<- of your hand->S<- transforms into a claw, which you can use as a weapon ->if it’s empty.<-

So if you're are holding something in your "other hand" EACH handS can't transfor because they count as a simple melee weapon. You can't equip a single melee weapon if you are holding a shield or other weapon in your hand

Make changes as you like and talk to your DM if it's silly. But that's RAW ✌🏻

(this also mean you can't, RAW, interact with anything else, like potions for example, because your hans are technically holding simple melee weapons.

Quote from SavageMerlin >>

The transformation only happens if both your hands are free, it says;

 

Claws:

->EACH<- of your hand->S<- transforms into a claw, which you can use as a weapon ->if it’s empty.<-

 

So if you're are holding something in your "other hand" EACH handS can't transfor because they count as a simple melee weapon. You can't equip a single melee weapon if you are holding a shield or other weapon in your hand

I'd say that you are reading it a bit wrong. Nothing in the text indicates that the transformation is stopped by the hand being not empty, it is just the ability to use it as a weapon (i.e attack with it) that requires the hand to be empty.