If it were me, I'd use a bleach solution strong enough, so that I wouldn't wonder if it was strong enough. I'd remove the filter, since it is used for water that will be consumed. Then I'd put solution in the tank(s) and drive a bumpy road if possible to agitate it. An alternative is letting it sit overnight. Then flush and rinse, and agitate. The last step would be a new filter, every season.
I have some experience with filters, and they are tricky. If they work, they trap the bad stuff. The problem is next use, there is a tendency for them to flush out some of the bad stuff into the initial flow. The output could be worse than unfiltered, because the filter becomes a breeding ground. This is more likely with common charcoal and paper filters, than a reverse osmosis membrane.
Imagine the science project occuring inside a Brita type filter.
I drink tap water here in Denver, but I can appreciate carrying water jugs for drinking on the road. Someone said alcohol kills germs.