dnd 5e – Can you wield a Greataxe and Claws with the UA Barbarian Path of the Beast?

The UA Barbarian Path of the Beast has the following option when you rage:

Claws. Your hands transform into claws, which deal 1d6 slashing damage on a hit. When you take the Attack action on your turn and make an attack with your claws, you can make one additional attack using your claws as part of the same action.

Can a level 5 Barbarian attack with a Greataxe or other two-handed weapon then attack twice with their claws?

This would look like this:

—- Take Attack Action —-
attack with Greataxe (1d12+2+STR)
attack with Claw (1d6+2+STR)
attack with Claw (1d6+2+STR)

On one hand it seems that it is allowed by the rules considering nothing is mentioned about only attacking with the claws or that the claws cannot hold a weapon.

On the other hand it seems odd to make two attacks with claws while holding something.

On the third hand (which may or may not be required) the feature makes no mention of the claw attacks needing to be from different hands.