The adviser says he takes "full responsibility" for advising Sir Keir Starmer to make the peer US ambassador.
Starmer's chief of staff departs at a time the PM is isolated and in potential peril.
Under official guidance, trade envoys have a duty of confidentiality over sensitive, commercial, or political information.
No 10 sources have called for Mandelson to return the money or give it to a charity that supports victims.
The prime minister said "outdated assumptions about how to make it into a successful career" have held young people back.