I see only one letter written before the coup, and it makes no mention of Pinochet or his alleged desire to protect Chile's constitution.
Seeing as Pinochet wasn't named commander in chief until after July 6th, it is unreasonable to expect that Frei would have mentioned him on July 6th. What Frei did say then, however, was this:
"There is nothing that can be done by myself, by Congress, or by any civilian. This problem can only be fixed with arms. I share your apprehensions, and I advise you to state them plainly to the commander in chief of the Armed Forces, hopefully today."
Pinochet was named commander in chief a few weeks later, thus becoming the officer that Frei had endorsed to depose Allende.