I had a similar idea along those lines, but borrowing elements of her introduction to Logan in Uncanny, pieces of Target X, and Not Forgotten, and hints at NYX. IE, Laura comes to Logan's and Xavier's attention when a notorious New York gangster and drug and sex trafficker Zebra Daddy is murdered with "parallel knife wounds." After discovering her identity during a brawl with Kimura while the two are investigating the killing, they learn the details of the project that created her from her belongings (including Sarah's illegible letter, which Beast is able to reconstruct). Logan catches her scent and realizes she followed him back to the school after their first encounter, so heads out alone to draw her into the open. She attacks him, but is able to talk her down from killing him and herself by revealing he knows what happened to her, and gives her the reconstructed letter so she can read her mother's words for herself. He asks her to come back to the school with him and promises to help her, but they're attacked by Kimura, who manages to subdue them both.
They're taken to a Facility installation, where Laura is tortured. Logan meets Adam Harkins, who positively geeks out over the capture of Weapon X. Logan manages to break free, however, kills Harkins, and rescues Laura from her cell. Laura insists that the base and any materials associated with them must be destroyed, to which he agrees. Harkins is killed during the rampage, and Kimura arms the guards with carbonadium rounds (introduce these in a scene earlier in the movie) and orders them to kill the pair. Laura and Logan are separated (or Laura sneaks away on her own), and Laura ends up confronted by Kimura. They fight and Laura is defeated, but not before explosives she and Logan set go off, forcing Kimura to flee. Laura is incapacitated, but Logan rescues her, and carries her out of the bunker before it goes up.
However he's pursued by the remnants of the base's security and is shot several times. He nonetheless manages to fight most of them off despite his wounds until Kimura arrives. She unloads a magazine of carbonadium into him before he finally goes down. Laura wakes up in time to see this, and charges her, but Kimura reloads and shoots her, as well. However before she can finish her off, Xavier arrives with the rest of the X-Men, having finally located them on Cerebro (have scenes after Logan's capture where they try to locate him and can't), and freezes Kimura in her tracks. He learns of her own past, and mentally orders her to finish off the remnants of the Facility, sending her on her way. Meanwhile, Laura crawls to Logan, who is being tended to by Hank, while the rest of the X-Men look on. Logan begs her not to give up on her own life, and promises that the X-Men can help her, before dying.
Laura is taken back to the school where her gunshot is tended by Beast until her healing abilities are able to kick in again, and is introduced to the students as Logan's daughter. Xavier discusses Logan with the rest of the X-Men, and the movie ends on a shot of Laura sitting in a classroom during a lecture (other students could be Rockslide, Mercury, Hellion, Pixie, etc.), while Hugh Jackman recites Xavier's "Mutation" monologue from the first film.
A major theme of the film could be Logan feeling out of place in the new timeline. He's been a soldier all his life, and in the piece achieved in this new future, now feels useless without a war to fight.