Blackstone did invoke the natural law. England operated in an Anglican context and that church incorporated the natural law in its teachings. They inherited the natural law from their Roman Catholic roots. Richard Hooker was the preeminent Anglican natural law scholar whose work would have been most authoritative for traditional Anglicans (even John Locke nodded his cap to wise Hooker).
However, Blackstone was an Tory who argued for the doctrine of absolute supremacy of the law of England. Of Parliament's power, he famously noted:
It can, in short, do every thing that is not naturally impossible; and therefore some have not scrupled to call it's power, by a figure rather too bold, the omnipotence of parliament. True it is, that what they do, no authority upon earth can undo.
As Gary North acutely observed: "Blackstone was wrong: beginning eleven years later, the American colonies undid a lot of what Parliament had done."
This thorny dilemma persists today. Scholars disagree on 1. whether natural law exists and should inform the content of positive law; 2. whether the principles of the Declaration of Independence accurately represent traditional natural law, or whether they are something more enlightened and modern; and 3. whether the natural law of the Declaration of Independence is "justiciable."
Time forbids me from discussing all three in this post. But let us focus on 3. Justice Scalia, because he rejects the Declaration of Independence as justiciable in American law has been hammered by Harry Jaffa and his followers for being a "legal positivist." Scalia is a devout Roman Catholic and I've seen him on record, unsurprisingly, claiming to believe in the natural law (which devout Roman Catholics do). Scalia need not answer whether he believes the Declaration of Independence's natural law accords with that recognized by the Church; American courts, according to Scalia, have no business using any conception of the natural law to decide cases and controversies or nullify actions of other branches of government. If natural law is to inform policy issues, that's the legislature's job. (And among conservative Catholic jurists, Scalia is by no means alone in his approach.)
And that's because some legal body must have the final say over how to interpret and implement natural law. Under a Blackstonian framework, it was Parliament. So it could be that 1. Hamilton was just wrong; what Great Britain did, did not violate the natural law and America had no business on natural legal grounds to disobey. OR, 2. even if Hamilton et al. were right, they still had no business disobeying British rule because some legal body has to have the final say over how to implement the natural law into governing law and under Blackstone's conception, again, it was Parliament.
Perhaps this is why Hamilton cites other natural law thinkers who may not have viewed things exactly as Blackstone did. This is not to say I have a problem with America's Founders clever, revisionist use of Blackstone this way. But let's see it for what it is: They took his principles, tweaked them a bit, and applied them to achieve results that Blackstone would neither have expected or approved of. This is Whig history 101.