Once I use the auto-hiding dock on the right screen, I am no longer able to use the dock on center screen. To resurrect the dock on the center screen I first have to use the dock on the laptop. Once I do that, the dock on the center screen magically comes back to life. This cycle is repeatable, without fail.
Here is my monitor arrangement. It has the top of the laptop just slightly under the bottom left corner of center monitor. This depicts the actual physical configuration.
Best Answer
While documenting this problem, I stumbled upon the solution -- after two years of fiddling with this. Note the new placement of the laptop in the arrangement below when compared to that in the original post. I had tried swapping monitors, cables, ports on my MBP dock, the menubar location. Nothing helped. And while I was writing up the original post I thought, I wonder if the laptop being at all under the center monitor makes the OS treat the two as one, vertical viewport when it comes to deciding to raise dock. Now, I don't know if that's accurate or not, but it got me to try moving the arrangement to have the laptop just around the corner on the side, and voila! No more strange dock behavior. Now I can freely use the dock on any monitor and it does not get stuck. Now, I just have to undo all the strange muscle memory my mouse arm learned while this was still a problem...
If anyone can speak to my supposition about the "one, vertical viewport", I'd be interested and grateful to know the details.