It is interesting that the soap is alkaline, and Lemishine is acetic acid, and therefore acidic (on the opposite end of the scale). Now acetic acid is a weak acid granted, but it is an acid nonetheless.
No, LemShine is Citric acid, Vinegar is a acetic acid. But either or they are both weak acids so yes the OP not getting good results with the Lem shine may simply be not using enough of it.