Clearly this is not a linear problem. Not an inability to provide sufficient fuel at peak demand but somewhere something is not providing correct fuel or ignition or air under what should be an easy state to maintain.
The problem has been ongoing for over a year unsolved? Its time to bring a shotgun to the party. Buying and installing a new fuel pump would be costly but swapping a gas tank only requires friendship, trust, beer+pizza, and perhaps 30 minutes. I'd swap gas tanks with someone long before I'd let you borrow my TPS, APS, or ECU.
Plus, if the problem moves to the other FJR you really know you have found it.
And problems tend to hide where everyone knows simply can not be the source of the problem.
In my career I have come across situations where oscillations appeared where one would think should have been the easiest conditions to regulate. Stable and steady is pretty hard because only small corrections are desired. Acceleration and deceleration are easy because you accept whatever rate of change you get.