If it were I, I would skip the filter and boil the water for 10 mins. Let it cool and then refrigerate it. Countertop for a couple days might be ok deepening on ambient temperature of your room. In other words, if your room is hot, the water might go bad more quickly than in the refrigerator. I store my boiled water in a glass pitcher at this time of year in the fridge.

I also have a Life Straw pitcher, which has a filter pore size of 0.2 micron, small enough to filter out NTM. Most of the year it's on the countertop because I finish what I fill everyday. I am not familiar with PUR.

In addition to managing GERD with products sleeping at an angle/elevation helps keep stomach contents from reaching the lungs.