Does this reference exist for Longines' vintage offerings?

I think that a few factors are pertinent here. Firstly, as @BlueHands mentioned, there are so many Longines references that I would not always expect to find a second example online. Secondly, Longines watches that were made in the mid-1940s and earlier did not have reference numbers on their case-backs. Up until fairly recently, this made it impractical to describe a watch by its reference number since few people would be familiar with specific references. Further, one would likely need to contact Longines for an extract in order to know the reference number, and not everyone is aware or interested in doing this. As a side not, people used to mistakenly call the 5-digit order numbers reference numbers. Thirdly, certain references are very uncommon so finding another example can be tricky. If you would be willing to share more information about this watch (e.g. the caliber, date of original invoice, photos, etc.), I would be curious.

P.S. Longines should have a record/cliche of the reference, so I would ask them to confirm that the case design matches their records.

Are you sure it was 5418? I am aware of a 5413 for sure.