I have not used the B&S kit as I make things from scratch generally following the instructions in Christina Anderson's book with excellent results, but here are a few thoughts...
I doubt that the paper is an issue. I use HPR as my main paper for salted-paper and have not had issues with it.
The processing described in the B&S instructions is rather simple compared to Anderson's method (wash 2x with 5% salt, 1x water, 2x alkaline fixer, 1x water and 1x hypo clear; 4 min in each tray with intermittent agitation.) This is followed by a final wash. You might consider trying this method.
Regarding your tap water... are you on municipal water or a private well? Municipal water is generally very consistent in its properties as it is monitored and adjusted before distribution. Water from a private well can be quite variable over time, often varying with the seasons.
I'm on a private well and the hardiness of my water increases significantly over the summer especially in drier than average years. I have not seen this affect my salted-paper prints, but it does affect my cyanotypes which are sensitive to alkalinity.