The question is what does Dawkins think science is if it isn't a social construct.
If I was Dawkins I would steer clear of the truthsayer role. I may share his general opinion about philosophy and religion but I certainly don't share his philosophy of science.
I understand the temptation to counter the cult of truths that is represented by almost every philosophy or religion with truth claims about science but it has the practical effect of creating new cultish thought patterns. Science is about probable facts not truths. Once you go down the truth path you are playing the same game as philosophy and religion.
When Dawkins says science what he means is unavoidably tied to institutions that are clearly social constructs. Institutions that come with all the flaws inherent in human institutions. All the same instinctual baggage. Science institutionally is a prestige economy. It's hierarchies of competence perhaps more empirical than most but still subject to corruption. One of those corruptions is taking on the truthsayer role. To use science in the abstract to raise your own prestige in the realm of politics. Politics corrupts everything it touches including Dawkins.
Science is inherently a humble business. It removes the subjective, the scientist, as much as possible from the equation. Once a scientist injects themself into anything they are no longer doing science. The scientist is a social construct.
Science is a tool not the product. The structure it creates inherently less perfect because of complexity than the tool itself. The edifice science builds is sturdy but it has to be constantly repaired and updated.
Confusing science with what it creates is an easy mistake to make. It is one of the failings of the philosophy of science that is prevalent.