So here's the thing...
No one likes a whiner; I get that. But to peel another layer of the onion, there are apparently 2 versions of the 2002 153IR. From your continuity measurements, it appears the 2002 versions may indeed be different in their wiring (based on measurements of a 2002 version in a previous OGR thread)....hence why I asked you to double-check.
I am slowly trying to unravel this insanity!
Bottom line is Lionel has changed the wiring instructions for the 153IR which impacts your ability to mix versions in the method I proposed. They did so while maintaining the same SKU or part number. And to my knowledge there is no "official" method to determine which version or revision you have. So far, and as I've instructed, the best clue is the labeling of the two power terminals (either U-A, A-U, or unmarked). But this is just my guess. And wouldn't you know it, apparently yet another version of the 153IR came out last year which for all I know will have yet different wiring instructions and another revision date! It's enough to make a grown man cry! 
As to using the "oddball" 153IR in the middle section...
You still need to use the timer module. Even though the direction does not change in the middle, the key feature of the timer module (when set to mode 1.1) is it performs its time-delay function irrespective of whether the trolley is still at the station. In other words it starts the time delay function when the trolley arrives at the station. A 153IR does not start its time-delay until the trolley leaves the station (moves away from the 153IR). So without the timer module, the 153IR in the middle section would kill track power when the trolley arrives and simple keep track power removed forever (never starting its timer).
