How can silver work as a disinfectant in a bathtub if precipitates in the presence of chloride?
If you're referring to chlorination of public water supplies, keep in mind that this is likely not the main source of chloride ion. Elemental chlorine disproportionates via the following reaction:

$\ce{Cl2 + H2O &lt;=&gt; HCl + HOCl}$

Most chloride ion probably derives from various natural and industrial sources, as well as miscellaneous human activity. One source (somewhat outdated) I found indicates an average chloride ion concentration in U.S. public water of $11.5$ mg/L, which equates to a molarity of $3.24 \cdot 10^{-4}$. Using the $K_{sp}$ of $AgCl$ you can calculate the concentration of silver ion necessary to produce a precipitate, but given that it has a value of $1.77 \cdot 10^{-10}$ under standard conditions, you can assume that a fairly minimal amount of silver would be required, even with the large volume of a tub and increasing solubility in hot water. It's likely the case that any precipitate is just finely dispersed in solution and not clearly visible.