I think You are mixing apples and oranges. The two are completely unrelated.
The way you phrase the question doesn’t make sense “From a halakhic perspective, do you consider the Constitution’s and Declaration of Independence’s assertion that the “rights” it enumerates are inalienable because they exist in natural law and were established by G-d”
Halachically there is no right to free speech , quite the opposite certain types of speech are explicitly forbidden. Halacha certainly does not support free exercise of religion.
The US constitution is made by man. We (ie the authors/society) felt that we would be better guaranteeing free speech. So we did, there is no halachic support for free speech (the opposite is true). Furthermore IT is quite possible that one day it will be amended and Free speech will be limited, perhaps by a halachically inclined group (I’m sure some coffeeroom posters would support an “except lashon hara” clause to the right of free speech.
The idea of “inalienable rights” is a legal fiction. In many western countries there ARE limits on Free speech, hate speech is not protected. Are they violating these “inalienable rights”? Many first amendment purists would say yes they are. Those countries would argue no they are not. Can you prOVE that G-d gave us the right to say whatever we want. OF course not, in fact depending on which Releigion you practice I can prove (or at least argue) the opposite
I defer to the lawyers on this, But I think even in American law it i’snt really true that there are “inalienable rights” ie rights that can’t be taken away. I think if the constitution is amended they can be. Granted this is intentionally a complicated task. But I believe (correct me If I’m wrong )that theoretically the right to say free speech can be removed under the amendment process.