Find Antony and agree to fight a few of his dogs in the Legion arena in exchange for the bear. This can be avoided if you have good reputation with the Legion or the Animal Friend perk. You may also pickpocket the bear from Antony or kill him for it.
Tear the teddy bear in half in front of her, resulting in bad Karma and Legion fame.
Behind the scenes
The name of this quest is a reference to the film Saving Private Ryan.
This quest is very similar to the quest Find Mr. Nixon in Fallout 2, where you find a child's doll and have the option to brutalize it directly in front of the child in question.
Bugs
Xbox 360 If you make a fight at the fort, occasionally Melody will flee to the fort's entrance. Along with all other Characters, not including the Courier, she can use an invisible door and end up on the other side of the gate. The other side of the gate is in the same cell as the Fort's interior and the blip will appear on the compass. While the npc is there you can see them by looking through two holes in the walls on either side of the gate. [verified]
Xbox 360 To fix this jump on the boxes next to Antony and Lupa (if still alive), then on two wall parts on the right, and try to jump over the wall. After that go to drawbridge gate, jump on the wall parts to pass through the rock. Use 3rd person camera to use drawbridge gate, so you'll enter Caesar's part of the Fort again (you are there already). Now Melody should be in the brahmin pen.