Think of the climb angle as the flight path vector. Angle between the horizon and the flight path of the aircraft.
Pitch angle is the angle between the horizon and the aircraft attitude.
The difference between the two is, for all intents and purposes, aircraft angle of attack.
AoA wil remain constant at constant IAS. Climb angle, as you state, will reduce at higher altitudes.
If that helps you should be able to work it out a bit more easily.