The DOI is a legal document that presents the Christian case for interposition. Up until Parliment [sic] attempted to nullify their rights as Englishmen the representatives of the colonies argued from their rights as Englishmen. When those were taken away they appealed to natural rights.
While one perhaps could make what the American Founders DID "fit" with interposition, one cannot make what they said in the DOI fit. For those unaware, John Calvin in his Institutes on the Christian Religion Book 4, Chapter 20, dealt with the Bible (mainly Romans 13, but other verses and chapters as well) Christians and government. There Calvin strictly interprets Romans 13 to argue submission to government is categorical because God demands it. Obedience to government is conditioned on rulers not making men affirmatively or by omission sin. So for instance, if government tells a Christian not to exceed 55mph you obey simply because government said so. If government says stop preaching the Gospel, you disobey but submit to their lawful authority when they come to arrest you and accept the civil legitimacy of the punishment no matter what it is. I know this sounds harsh, but it is what John Calvin taught and Calvin was a harsh dude. [For more see Dr. Gregg Frazer's article on the matter.]
To Calvin, the Bible categorically forbids revolt. No exceptions. Calvin did discuss the ability of intermediate magistrates to interpose and remove a tyrannical King; but he stressed it must be done pursuant to some positive legal mechanism, like the Congress impeaching the President pursuant to the provisions in the US Constitution. Again, revolt is still forbidden. Therefore if the Continental Congress could make the argument, which they seemingly did in parts of the DOI, that King George and Parliament were violating British law AND if there were some recognized legal method under British law for declaring independence, perhaps what the FFs did could "fit" with such a notion of "interposition."
Needless to say, William Blackstone, the recognized authority over British Common Law argued there was no lawful right whatsoever to revolt against the Parliament and King like America later would do. The DOI is an anti-Blackstonian document in that respect.
But even if it were possible to make what America did "fit" with Calvin's notion of interposition, it is impossible to make what they said in the DOI fit. That document states:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. — That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, — That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it,...
Sorry but this is a call for revolt against the positive law, not interposition by lawful magistrates consistent with governing positive law. The above quoted rhetoric says the opposite of what Romans 13 seemingly on its face says and is entirely foreign to John Calvin's interpretation of Romans 13.