Forgetting for a moment that I value trees...
Lets look at this:
"The engineer thinks the trees are drawing too much moisture away from the ground, thus causing the movement of the basement wall"
No question about it that shrink/swell of soil causes damage to basement walls. I'm not in Kansas (anymore), but around here, it isn't the 'shrink' part of that equation that anybody worries about. It is the swelling that has pushing power.
Secondly, is the suggestion that if trees are gone, the soil against the foundation will not dry down as much? How strong of a certainty is that?
Third, how much MORE water will be added to the soil because they remove 3 giant water pumps (the trees...) from the system. This seems to have the potential to create more swell leading to more significant volume change when (not if) soil does dry down and shrink.
Finally, if the problem is things getting too dry, wouldn't a simple soaker hose on a timer solve that for hundreds of dollars (when you include the cost of the water) compared to thousands of dollars for tree removals.
Again...around here, the problem isn't vegetation. It is back-fill with clay rather than gravel and it is water not being diverted away from the foundation well.