I can tell you what I would do- I’d disinfect it in place. Those things are designed last to fit in between everything else and not easy to remove. This will clean the bottle and lines.
My vehicles can blow through washer fluid in no time so I wouldn’t worry about the pump. That being said, they’re not meant for continuous duty so I’d probably run it periodically with some time to rest in-between. First, I’d disconnect the outlet line at the most convenient location (bottle, T or at the wipers for example) and put it into a suitable sized container. Flush all the old fluid out into the container. Smell it. Does it smell bad? Yes, you’re on the right track. No, time to look elsewhere.
Now mix up a batch of disinfecting concentration of bleach and water. Fill reservior and let sit for 30 minutes. Now flush it out. Repeat as necessary and then flush with clear water. Fill with washer fluid.