Brita filters only improve smell and flavor, they don't filter or kill germs. I think the issue with your cooler may be that if not enough water is run through it every day, the water inside will stale and smell and taste bad. Also, the filter probably also should be replaced and some descaler run through the machine. At least that's whats done on plumbed in coffee machines we have here.
I'm not sure what the concern is regarding getting water from the bathroom. It's supplied from the same pipes as the kitchen water and in our case, the bathroom has more water usage and is cleaned more often than the kitchen! I personally don't like this practice, but many people wash their cups and stuff in the bathroom instead of the kitchen and they aren't always getting sick.
As far as bacteria breeding in plastic bottles, i think these are just stories that the 11 o'clock news likes to carry because they have an 'ick' factor that raises ratings and gets people talking. They never actually go into what kind of bacteria are bred and whether they are in fact harmful and how many are needed to cause illness etc. And they usually have some droid from the 'bottled water marketing council' talk about how bad it is for you, never mind how bad it is for their profits if people just refill a reusable bottle.
For that matter, if this is such a problem, why are open Sparkletts trucks allowed to drive around and deliver water all day in the sun? Wouldn't that be a bigger source of contamination and bacteria growth?