In the existing route I satisfy Your Father's bones: a Sky-light for the Fathomking by bringing his Complexity a sunlight-filled mirrorcatch box. However the fastest way to acquire an item for this part of the ambition is recruiting the Presbyterate Adventuress and giving her up in the Boarding Action event in London, which will reward the player with a Mountain Sherd. This does add more officer rng to the run though, so some runners may wish to stick with the Aestival strat as they're starting out.

It is worth noting that Priest is actually slower than Soldier for this ambition. Soldier can advance the ambition by compiling port reports at Hunter's Keep, Kingeater's Castle, and Empire of Hands then talking to the Dark-Spectacled Admiral. Soldier can then fast travel directly to Kingeater's Castle from Fathomking's hold at the end of the run and then from Kingeater's directly to London. The downside of Soldier is that a lower hearts attribute will make your crew more vulnerable to fire checks and your runs will die to rng much more often. You will also have to more careful about conserving fuel at the beginning of the second loop since you can't get fuel at Kingeater's with the gall-eyed engineer, and you will need to be more cognizant of where your secret is coming from for the fast travel back to Kingeater's from Fallen London, since you aren't perusing the EoH port report in your study. It's also worth noting that the EoH port report is worth 300 echoes, so it might mean being able to cut out the first three captains thus saving about a minute and a half, but I'm not sure.

It is at least conceivably possible to do this run without completing the neathbow, but the route would require getting the nacreous survivor from the Fathomking, which is prohibitively expensive and may require too many neathbow colors to save time at all.