  The correspondence between Gorenstein Fano toric varieties and reflexive
polytopes has been generalized by Ilten and S\"u{\ss} to a correspondence
between Gorenstein Fano complexity-one $T$-varieties and Fano divisorial
polytopes. Motivated by the finiteness of reflexive polytopes in fixed
dimension, we show that over a fixed base polytope, there are only finitely
many Fano divisorial polytopes, up to equivalence. We classify two-dimensional
Fano divisorial polytopes, recovering Huggenberger's classification of
Gorenstein del Pezzo $\mathbb{K}^*$-surfaces. Furthermore, we show that any
three-dimensional Fano divisorial polytope is equivalent to one involving only
eight functions.
