Specific errors
I also remember at the beginning of my training being told that the compass is only accurate in "straight and level, unaccelerated flight".
Subsequent training for both CAA and FAA Cpls, IRs and Instructor ratings has then specified both turning errors and acceleration/deceleration errors associated with the compass but never an error caused solely by pitch, unless there is an associated acceleration or deceleration.
From a practical point of view it should not be necessary to align the Heading Indicator during the climb if you planned things properly prior to that, however I do not of a reason why for example if you are on partial panel in a straight constant speed climb that you should not be able to rely on the magnetic compass to confirm your heading.
I will ask ground school and see how many different answers I can get.
Okay first answer already.
No errors in a straight unaccelerated climb related to tilting of the bowl.
However due to the change in relative position of other components of the aircraft and engine in the climb attitude relative to the compass, and therefore changes in the three components of the magnetic fields which cause deviation there could be an overall change in compass deviation.