Bearing in mind that the phrase "comes to the pinch" means "when life's circumstances become problematic", I find it hard to imagine a situation in which that might be applied to claws! "When it comes to the pinch, claws........" Well, what?
On the other hand, I can imagine David Attenborough saying "When it comes to the pinch, crabs usually manage to survive polluted water", for example.
The word 'they' in the clue definitely suggests to me a living creature rather than a body-part.
But what the hey! You choose whichever seems the more likely to you, Pollyanna.