No. While the words "due process" appear in both places and absent specific internal language that they were actually copied from the first place, they first have to be given their meanings according to that understood at the time they were used. To assume they were "copied" means that judicial construction is already being applied before a determination of ambiguity, which enables construction to be used. A Catch 22? No. Construction cannot be applied until after ambiguity is found de novo.
Well, here's where I'd have to disagree. For one thing, there's no reason why the 1868 understanding of "due process" should be any different from the 1789 understanding in the first place. Secondly, any law should ideally require no inquiry into original intent, because the intent should be stated in the law, or at least, the intent should be discernible by examining the text of the law. Of course, I know that it's unrealistic to expect that it will always work out that way, but that should always be the goal in writing down any law - indeed, in writing down just about anything at all, but law especially. No drafter should ever plan to have people wonder about his intent or understanding. To me, that completely defeats the purpose of having written law in the first place. And so the Constitution should read as though it is one document, from start to finish. Which mean that words and phrases that appear in one part of the document should have the same meaning in another (unless the context is clearly different). Which means that any reasonable person picking it up and looking through it should have a reasonably good chance of being able understand what's being said, without constantly having to go, "OK, when was this part written? When was it ratified? What was going on at the time? etc." Like I said, there are times when he may have to do just that, but that should be the exception, rather than the rule.