No, not as written
Two weapon fighting requires you to be holding two different light weapons, unless you have the dual wielder feat (which removes the light requirement).
Your claws are arguably not being held, and definitely do not have the light property.
(I would advise against taking the dual wielder feat for this, as the feat isn't great even in optimal situations. And it your claws aren't an optimal weapon to dual wield with.)
It doesn't break much ... at level 5
By the rules as written, you can grab yourself a two handed sword. You can make one two handed sword attack, one claw attack, and then an extra claw attack.
The potential damage done is 4d6 plus 3x your strength modifier, plus 3x rage damage. With 18 strength this is an average of 14+12+6=32, and this does not require your bonus action.
In comparison, if you treated claws as being two weapons with the light property, you could do claw, claw, beast extra claw, bonus action claw for 4d6 plus 3 times your strength modifier, plus 4x rage damage. With 18 strength this is an average of 14+12+8=34, minus 5.5 if you don't have a bonus action.
If you treated claws as being a single weapon, but held and with the light property, you could wield a short sword. Then you could do claw, sword, beast action claw, bonus action sword, for a total of 4d6 plus 3 times your strength modifier. This does the same damage as the above.
(In comparsion, a baseline 2H sword barbarian does 4d6+str x2+rage x2, or 26, and a 16 strength reckless GWM does 44 at -6 accuracy.)
At level 3
If you permitted claws to be 2 different light held weapons, at level 3 you could do claw, beast claw, bonus claw, dealing 3d6+2x strength+3x rage damage (22.5).
If you treated it as a light weapon held, but only one of them, then claw, beast claw, bonus short sword deals the same damage.
Without treating them as light, we deal 2d6+2x strength+2x rage (7+6+4=17).
(In comparison, a baseline 2H sword barbarian does 2d6+strength+rage or 13.)
Balance impact
Reading the claws as light and held (in the hand that is free) is only a 6% increase in damage output on turns you have a bonus action free, and a reduction in damage output on turns you don't have a bonus action free of 11%. As you lack a bonus action the turn you rage, this is pretty much a wash.
At level 3, however, this technique allows a 32% increase in damage output.
What more, Path of the Beast (Claw) is already a decent damage barbarian build; it incompatible and viable competition to a GWM barbarian; the PAM+GWM VHuman is clearly higher (58.5 average damage at -6 to hit is worth it with reckless). And at level 3 it is by far one of the higher damage builds.
What I'd do as a DM
By RAW, you can't do this.
But at level 5, allowing this trick isn't worth ruling against as a DM, as the visuals are more fun and the balance impact is zilch.
So what I would do is allow it, and require the separate weapon. Ie, your claws are wielded, but considered the "same" weapon, so they can't TWF by themselves.
I would also only allow it at level 5 and above.