Why is the 737’s aileron/spoiler authority reduced at low flap settings?
Okay, turns out this wasn't so different from the crossover-airspeed question after all!
To quote @PeterKämpf's answer to that question:


how can [an airplane’s ailerons and spoilers] lateral control authority surpass that of the rudder past a certain speed?

This depends on the wing's lift coefficient. At a higher lift coefficient the lower aileron can not add the same amount of lift that it could at a lower lift coefficient. While the raised aileron on the opposite side will still reduce lift locally, the lowered aileron becomes less effective in raising lift as the lift coefficient increases. A higher lift coefficient causes a higher suction peak near the leading edge and puts more stress on the boundary layer, and adding more of the same will become harder as the wing approaches stall conditions.
Another factor is adverse yaw which increases with lift coefficient. This adverse yaw will add to the yawing moment of the hard-over rudder and will increase sideslip, which in turn will produce more rolling moment from dihedral against the aileron effect. As adverse yaw dies down with lower lift coefficient, so does sideslip angle and the ailerons gain control power.

Which also explains why the 737's crossover airspeed is lower with flaps down; when the flaps are up, the aircraft has to fly at a higher angle of attack in order to generate the same amount of lift at the same speed.  This increases the lift coefficient over the entire wing, including the outboard portion where the ailerons are mounted, which makes the downwards-deflected aileron, to use Peter Kämpf's terminology, &quot;less effective in raising lift&quot;, and increases the adverse yaw generated by the deflected ailerons.  All of which decreases the roll authority of the ailerons, and, since the spoilers' roll authority can't increase to compensate, decreases the roll authority of the aircraft's lateral controls as a whole.
With the flaps down, on the other hand, the lift coefficient of the inboard wing increases, which means that the rest of the wing doesn't have to generate as much lift, which means that the aircraft can fly at a lower angle of attack for a given speed, which decreases the lift coefficient of the outer, flapless, aileron-containing section of the wing, which increases the effectiveness of the lowered aileron and decreases the adverse yaw generated by the ailerons, which increases the aircraft's roll authority.