I have a wire brush designed for cleaning out the tube on a Platypus water reservoir that goes into my backpack and comes out to a bite valve on the shoulder strap of the backpack. I think it is about 2.5 or 3 feet long. You can buy the cleaning brush at REI or some other sports store that sells hiking gear. You could go from each end to the middle and then flush the tube out.
I also like the idea of liquid dishwasher machine soap. I have used this for cleaning many stubborn things.I used some in the black tank of the truck camper just after we got it.
It came from a rental fleet so tank flushing may not have been done properly. I had flushed the tank and thought all was good , but added some Cascade into the black tank, added 5 gallons of water and drove around for 2 or 3 days. The water that came out showed me that the tank was not rinsed during its rental days.
Could also try to simply reroute a new line from the pump to the tap.