The White House is considering a bid for the late MP to receive America's highest civilian honour.
Sir David Amess's daughter says the family will engage with the independent review in "good faith".
Katie Amess says previous reviews have failed to provide answers, and she wants a full inquiry.
The Home Office says it will appoint a key figure to scrutinise past reviews for Sir David's family.
A major review found the counter-terror scheme could have potentially turned the killer away from violence.
The family of the murdered MP say they still feel betrayed but are hopeful of answers.
The family of Sir David Amess say the decision is "unacceptable" and "insulting".
The programme's handling of the Conservative MP's murderer was "sub-optimal", a review says.
A former MP says the families of Sir David and the Southport victims have been equally failed.
