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Following recent claims relative to the question of large anisotropy production in regular bouncing scenarios, we study the evolution of such anisotropies in a model where an ekpyrotic phase of contraction is followed by domination of a Galileon-type Lagrangian which generates a non-singular bounce. We show that the anisotropies decrease during the phase of ekpyrotic contraction (as expected) and that they can be constrained to remain small during the non-singular bounce phase (a non-trivial result). Specifically,we derive the e-folding number of the phase of ekpyrotic contraction which leads to a present-day anisotropy in agreement with current observational bounds. [ABSTRACT FROM AUTHOR]