Biblical covenants involved a suzerain and a subject or subjects. For example, in the Abrahamic Covenant, God, the sovereign, grants Abraham, the subject, and his descendants, the land of his sojournings, to be possessed in what was then the future. In the Davidic Covenant, the same God grants that David's dynasty will last forever. In the Mosaic Covenant, God lays down the behavioral requirements for the Israelites to live securely in the land that was promised.
I see no similarity between the Biblical Covenants and the US constitution, which involves the people of the US securing and protecting their own individual rights through the power of a federal republican government. Our constitution restrains government. In the case of the Mosaic covenant, it restrains the people. The other covenants involve GRANTS from God to a person or people.
If one wants to find a comparison between Biblical covenants and human equivalents, one should look not to the US Constitution, but to the ancient Hittite suzerainty treaties.
All of which is to say, there is no sense in which our beloved United State is in covenant with God. The USoA is not "my people who are called by my name". Saying it doesn't make it so. There was only one typological Israelite theocracy.
< /soapbox>