[RPG] way to mix Unarmed Strike Damage with Natural Attack Damage or vice versa

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.

Delivering a spell as part of an unarmed strike

In answer to your question, yes a monk can deliver a touch attack spell through his normal attack. However you have to hit against their normal AC to do so, not their touch AC. If you miss their AC, you fail to do damage and the spell does not fire (even if you miss their AC, but hit their touch AC). If you hit, you do all the normal damage of your attack, and then the spell goes off. That's the advantage and disadvantage of doing it this way. The rules for this are here (thanks to @Tridus for the link).

Delivering a spell as a touch attack using natural weapons

A cleric/monk could also deliver the spell as a normal touch attack, rolling against the opponents touch AC, but doing no damage for his fists. This works no differently from any other touch attack because

1. using a touch attack counts as an armed attack
2. natural attacks count as being armed anyway

You can think of the difference between this and delivering it as part of a normal attack as a normal attack being a full on punch with a spell going off as the punch connects, whereas just going for a touch attack is just tapping the opponent anywhere, not bothering to penetrate their armour, and the spell goes off the moment you make contact with them.

Other classes

There are several other classes that do this type of thing by design, such as the Duskblade, who can deliver spells through their normal attacks. Having a look at the rules for them may give you further insight, if you need any.