P Factor is attempting to yaw the airplane to the left. If the climb is performed correctly (sufficient right rudder is applied to correct), the airplane will not yaw, or change heading. If there is no yaw or pitch change, there is no precession reaction. If the pilot is lazy and allows the airplane to turn, then precession occurs while the turn is happening.

Excerpts from the Pilot's Handbook of Aeronautical Knowledge again, page 3-24 and 3-25 ( http://www.faa.gov/library/manuals/a...ilot_handbook/ ):

"GYROSCOPIC ACTION
... Precession is the resultant action, or deflection, of a spinning rotor when a deflecting force is applied to its rim. As can be seen in figure 3-32, when a force is applied, the resulting force takes effect 90 degrees ahead of and in the direction of rotation.

...

It can be said that as a result of gyroscopic action--any yawing around the vertical axis results in a pitching moment, and any pitching around the lateral axis results in a yawing moment...."

Emphasis added.

The last sentence requires that the gyro is free to change orientation as a result of the force applied to it. In the case of a steady-state climb, yes, P Factor is working, but right rudder is also working to cancel it out, so the gyro is not free to change orientation, hence no precession.

Your statement that, "The plane of the prop isn't changing but the forces are still there." is incorrect IF you are referring to gyroscopic precession. P Factor and precession are different phenomena with different causes. By the above, definition, if the plane of the prop isn't changing orientation, then precession does not occur. In a steady-state climb (aircraft attitude is constant), the plane of the prop isn't changing, and the P Factor force is occurring, (although it's being compensated for with rudder so the airplane isn't yawing), and precession is not occurring at all.