Yes, the team is allowed to attach the Prosthesis to the Pet inside Base.
1) R10, Handling During the Match, disallows touching anything (except the robot) that is outside of Base. There is no rule that disallows handling objects that are completely inside Base (or Storage), so that is allowed (GP2).
2) Normally the team would not be allowed to attach the Prosthesis to the Pet (R11), --but-- Mission M11 requires it. By rule GP5, Information Superiority, the Missions can override the Rules, so attaching the Prosthesis is allowed.
As Todd noted, the Referee may take the Pet/Prosthesis combination out of play because of R14, Interrupting. If that happens, then the team will not score any points for M11.
Steve Scherr
Virginia-DC FLL Referee Advisor