Would lowering pH of the lactase solution affect the enzyme?

Lactase is made of proteins. Thus, changes in pH will change the tertiary structure of the protein, resulting in the active sites of lactase being uncomplementary to its substrate, thus denaturing it

yes its true that any change in ionic environment has the ability to change the tertiary structure, active site etc. of an enzyme and therefore limit its function.  Even in your example where this enzyme functions normally at low pH, lowering the pH even further will have the same effect and lessen its specific activity.  All enzymes have an "optimal" rate of activity at a specific pH, and this rate goes down below and above it.  If you've seen a graphic of this it looks like a hill, the rate of catalysis approaching zero far below and far above the optimal pH.