  Counts of galaxy clusters offer a high-precision probe of cosmology, but
control of systematic errors will determine the accuracy of this measurement.
Using Buzzard simulations, we quantify one such systematic, the triaxiality
distribution of clusters identified with the redMaPPer optical cluster finding
algorithm, which was used in the Dark Energy Survey Year-1 (DES Y1) cluster
cosmology analysis. We test whether redMaPPer selection biases the clusters'
shape and orientation and find that it only biases orientation, preferentially
selecting clusters with their major axes oriented along the line of sight.
Modeling the richness-mass relation as a log-linear relation, we find that the
log-richness amplitude $\ln(A)$ is boosted from the lowest to highest
orientation bin with a significance of $14\sigma$, while the orientation
dependence of the richness-mass slope and intrinsic scatter is minimal. We also
find that the weak lensing shear-profile ratios of cluster-associated dark
halos in different orientation bins resemble a "bottleneck" shape that can be
quantified with a Cauchy function. We test the correlation of orientation with
two other leading systematics in cluster cosmology -- miscentering and
projection -- and find a null correlation. Analytic templates for the
triaxiality bias of observed-richness and lensing profiles are mapped as
corrections to the observable of richness-binned lensing profiles for redMaPPer
clusters. The resulting mass bias confirms the DES Y1 finding that triaxiality
is a leading source of bias in cluster cosmology. However, the
richness-dependence of the bias confirms that triaxiality does not fully
resolve the tension at low-richness between DES Y1 cluster cosmology and other
probes. Our model can be used for quantifying the impact of triaxiality bias on
cosmological constraints for upcoming weak lensing surveys of galaxy clusters.
