How should multiclass classifier performance be measured when one type of error is preferred over another?
Define a scoring table like this (You will need to tweak this table to satisfy your particular use case. I am only using it as an example).

Pred   | True Label
Label  | A    B   C  
-------+--------------
A      |  0  -1  -2
B      | -3   0  -1
C      | -4  -3   0


Notice this scoring table has the property of favours over-estimate than under-estimates. 

Multiply (element-wise) the scoring table with the number of predictions fall into each of the 9 possible scenarios then sum the scores together will give you a metric with the desired properties.