  We produce the first examples relating non-orientable exact Lagrangian
fillings of Legendrian links to cluster theory, showing that the ungraded
augmentation variety of certain max-tb representatives of Legendrian $2$-bridge
links is isomorphic to a product of $A_n$-type cluster varieties. As part of
this construction, we describe a surjective map from the set of (possibly
non-orientable) exact Lagrangian fillings to cluster seeds, producing a product
of Catalan numbers of distinct fillings. We also relate the ruling
stratification of the ungraded augmentation variety to Lam and Speyer's
anticlique stratification of acyclic cluster varieties, showing that the two
coincide in this context. As a corollary, we apply a result of Rutherford to
show that the cluster-theoretic stratification encodes the information of the
lowest $a$-degree term of the Kauffman polynomial of the smooth isotopy class
of the $2$-bridge links we study.
