The 4th, 5th, and 6th Amendments should come into consideration immediately upon a receipt at State of any extradition request. The Secretary of State and all executive officers (and all Senators and Representatives) take an oath to support the Constitution (required by US Constitution, Article VI). The President takes an oath explicitly defined in Article II, Section 1, "to ... faithfully execute the office of President and ... preserve, protect and defend the Constitution of the United States."
No doubt Amanda and her lawyers have petitioned the Secretary to remind him of the violations of her rights, under Italian law and Constitution, and under ECHR case-law, by the Italian courts.
Futhermore, Senator Cantwell has stated that the trials were flawed. In the US, the government takes the statements of Senators quite seriously.