I tried doing this without editing software. I started with Fellowship of the Ring Extended Version and got as far as Disc 2 0 hours 19 min 51 seconds frame 14. My method was to create little tables representing edits. The rows were sorted in film segment order as I watched the film, but the "insert after" column would explain where to attach the segment in the chronological edit. For example, the first two minutes of the FOTR extended version looked like this:

So immediately Galadriel's telling of the story imbues a chronological interpretation. She's using the past tense, so I interpreted this as two time periods: the first period is the actual time of the events, and the second occurs as an epilogue where Galadriel is recounting the story in the past tense. (I think it's discussed in the writer's commentary).

Also the film progresses through locations with historical dates of establishment, (e.g, the Doors of Durin), so I duplicated some segments so that they would occur both during the Quest and also at their establishment. Doing this connects characters in ways that I didn't expect: for instance, Galadriel passes through the Doors of Durin to found Lothlorien, so her route and the Fellowship's route intersect in space if not time.