I have both found this bug and a means to bypass it for now, however onerous it is.

On disconnecting a LAN game from one computer to another and reconnecting, that user will come in "fresh", rather than their last known coordinates/inventory. After some sleuthing, I've discovered that the naming conventions for the player dat file have shifted from player name to what I'm guessing is session token. What was only 2 data files became 3 when the 2nd player reconnected.

The solution for now, that I've found, is to disconnect (but not exit Minecraft), copy the file name of the new dat file, delete it, then rename the dat file it should be to the file name that was copied. On reconnect, player will find themselves where they are supposed to be with the appropriate inventory. Each time the client is closed, it would appear a new dat file is created when joining the LAN session again, and the process must be done again (log in, log out, copy/delete/rename, log back in).

PS: Bug report would not allow me to select version 1.7.8, which is the version I'm using on both computers, which is the latest version at this time. Selected 1.7.9 even though that doesn't appear to exist yet.