I managed to tie it in fairly directly.
SPOILERS follow.
In my version, Lucan had, before his downfall, been involved in investigating the events surrounding Lady Elaydren's recovery of the first and second schemas. He interviewed her directly, found out about the involvement of the Order of the Emerald Claw, but ultimately decided this was a matter for House Cannith to police, not a concern for Brelish security.
Cut to when he's taken over by the intelligent sword, and it learns from his memories about the schemas, and that they're prized by the Order of the Emerald Claw, who are supposedly Karrnathi loyalists. Therefore these schemas must be of value to Karrnath, and it would be worth recovering one if the opportunity arises.
At around this time, an auction is being organised of the effects of a recently-deceased adventurer, and these effects include one of the schemas. The party read about this auction in the Sharn Inquisitive and recognise the schema, shortly before receiving a message from Lady Elaydren asking them to meet her at the auction house on the night of the auction.
The adventure opens in the auction house, at the top of one of the towers in the more opulent areas of Sharn, around two hours after sunset. The party get seats near the back, and watch (or participate if they like) as Lady Elaydren, seated near the front, bids upon the schema. Things go smoothly enough until a tall gentleman in the front row asks to examine the schema. As soon as he gets his hands on it, he knocks the auction house officials aside and leaps out of a tall window in the nearest wall. If the party hurry, they can just manage to catch a glimpse of him striding down the side of the tower before the bat swarms arrive to cover his escape and combat begins.
After the dust settles, Lady Elaydren tells them that she recognises the man as Lucan, a member of the Dark Lanterns, and offers them a reward to recover the schema for her. She sends them to see Viorr Maelak, head of the Dark Lanterns, and from there the adventure proceeds as written, except that when they finally capture or kill Lucan, they find the schema sewn into the fabric of his jacket.