Let k be a locally compact topological field of positive characteristic. Let L be a cocompact discrete additive subgroup of k. Let U be an open compact additive subgroup of k. Let ℓ, u and a be elements of k, with a nonzero. We study the behavior of the product as a varies, using tools from local class field theory and harmonic analysis. Typically ratios of such products occur as partial products grouped by degree for the infinite products representing special values of Gamma-functions for function fields. Our main result provides local confirmation for a two-variable refinement of the Stark conjecture in the function field case recently proposed by the author.