I would not expect some inverts to do well in tap water - there's SOO much crap in even the best tap - various metals, flouride, chloride, chloromine, silica, phosphates, salts, sediments (dirt), etc etc..... It's 95% possiblity there's somthing in your tap that some inverts are not ever gonna like, while others may not be affected by the contaminate as much.
Tap water quality can also vary ALOT druing different times of the year - depending on the chemicals needed to maintain what ever standards your city/community have. For the cost of killing of various inverts over time experimenting with what may be killing them in your tap water, and running the risk of having the city add something oneday that wipes out most of your tank or causes a crash - why don't you just get a simple RO/DI unit and be done with it