This gives a different angle of incidence on each wing,
Angle of incidence is fixed by the structure of the airframe - I sure hope left and right wings are the same!
No, because the aircraft is climbing
The angle of attack of the two wings will be slightly different in a turn as previously discussed, and I think agreed. The angle of attack might be affected by the climb of the aircraft. However, if there is a difference between the left and right wing's AoA as a result of the turn, that difference will remain the same difference despite a climb or descent as long as the radius of the turn remains the same. Based upon the power available, the aircraft could climb, fly level, or descend at the same pitch attitude relative to earth. This would result in a different whole aircraft AoA, based upon rate of climb or descent, but the difference left to right would remain the same, and thus the handling associated with that difference would remain the same in a climb, descent, or level.
Though aileron application can affect AoA, it's probably better for this discussion to exclude that factor, as it is certainly different for different aircraft, and there is no common rule there. My experience with the subtle changes in roll contorl and stability following the installation of a STOL kit leading edge cuff on a Cessna wing have shown me the affect on roll control, and dihedral effect made by changing and displacing a part of the leading edge radius downward.
The center of lift in a rotor disc, and the change in that center of lift postion based upon forward speed do bear some similarity to this discussion, but I think are different enough to not be effectively relevent to this discussion. I do agree that there are widely varied AoA's along the blade span though. Some blades have considerable washout to compensate for this.