A support for any book relating to any aspect of religion from Dawkins would be enough to persuade me not to bother reading it. The whole question of Buddhist teaching relating to karma and rebirth require a subtlety of thought quite beyond the average western-educated scientist. And quoting the Dalai Lama is another example of how some westerners, especially from the USA, naively believe that quotes from famous individuals somehow reinforce their own approach to the subject.
I wouldn't disagree with the quotation from Dalai Lama, but the whole aspect of karma relates to pratitya-samutpada and can't be regarded as something simple, nor does every Buddhist accept that the Buddhism of Tibet is the last word on the topic.
The dismissal of rebirth as 'arrant nonsense' may seem unexceptionable to the average westerner, but this attitude takes no cognisance of two facts: 1 - that Barash's western education and upbringing is within the context of a world view that dismisses 'reincarnation' (a term generally eschewed by Buddhists because of its implying an entity that incarnates). So by dismissing it out of hand he is merely staying in his own cultural rut and 2 – the Buddha clearly dismissed both a belief in everlasting life (eternalism) and belief that death is the absolute end (materialism) as both being incorrect views which should not be adhered to. If Barash wishes to be taken seriously he needs to ponder that problem, and think about the Buddha's advice to avoid clinging to a particular view.
Buddhists should be very wary of regarding the likes of Barash and Dawkins as supporters of the Buddhist view, because they are based on no substantive experience of practising it, and have no deep knowledge of Buddhist philosophy. Their ideas and views are of no significance.