###No one knows

The Beast Strike feat is poorly written. A good idea, but poorly written. Nothing says he actually uses his claws, and as you say, a Monk can make Unarmed Strike attacks without using his hands, which usually leaves hands free for Claw attacks. My guess would be that whoever wrote the feat, however, was completely unaware of this fact.

RAW, you can use the Claw attack(s) as secondary weapons in a full-attack after all normal attacks (provided you haven’t used your hands, which a Monk need not do), and a Flurry of Blows is explicitly a form of full-attack. This seems pretty likely unintended, but nothing can really be said with absolute certainty.

I probably wouldn’t allow it. I’d sooner fix or replace the Monk class than try to “throw him a bone” by allowing weird results from a poorly-written feat.

No one knows

The Beast Strike feat is poorly written. A good idea, but poorly written. Nothing says he actually uses his claws, and as you say, a Monk can make Unarmed Strike attacks without using his hands, which usually leaves hands free for Claw attacks. My guess would be that whoever wrote the feat, however, was completely unaware of this fact.

RAW, you can use the Claw attack(s) as secondary weapons in a full-attack after all normal attacks (provided you haven’t used your hands, which a Monk need not do), and a Flurry of Blows is explicitly a form of full-attack. This seems pretty likely unintended, but nothing can really be said with absolute certainty.

I probably wouldn’t allow it. I’d sooner fix or replace the Monk class than try to “throw him a bone” by allowing weird results from a poorly-written feat.