An old result of Müller and Rödl states that a countable graph $G$ has a subgraph whose vertices all have infinite degree if and only if for any vertex labeling of $G$ by positive integers, an infinite increasing path can be found. They asked whether an analogous equivalence holds for edge labelings, which Reiterman answered in the affirmative. Recently, Arman, Elliott, and Rödl extended this problem to linear $k$-uniform hypergraphs $H$ and generalized the original equivalence for vertex labelings. They asked whether Reiterman's result for edge labelings can similarly be extended. We confirm this for the case where $H$ admits only finitely many Berge cycles.