Adding the lime to boost the mash pH a bit is not mandatory, but it does help express the bitterness. If you are having difficulty getting your bittering to agree with the calculated level, then its possible that an overly low pH is working against you. Conversely, if the bitterness tends to be harsh and excessive, then the pH is probably too high.
In the case of the pale ale profile with its high Ca and Mg content, those ions depress mash pH through the Residual Alkalinity effect. The tap water has fairly low alkalinity and I'm not surprised that a bit more alkalinity is needed.
Producing a 5.3 pH isn't the end of the world. It will still be plenty drinkable. If the hops and bittering don't come through as well as you would like, do plan for a slightly higher mash pH to help.