Solution 1: Make a tool.
When I played the flute I made what amounted to a shepherds crook with an almost closed loop on one end. Thread the corner of a rag through it, and draw it through the body of the instrument.
You could do the same, but instead of 1/10" brass rod, you would use bailing wire, and instead of a cloth rag, I'd suggest either a scrap of sponge, or a scrap of 3M scotchbright pad.
Do note: Both of these materials will pick up dirt and crud from the tube, but will be difficult to get completely clean themselves. If you are only using water, this isn't a problem. If you use something with sugar in it, then there is the possibility that mold can grow on your cleaning apparatus. Rinsing thoroughly is in order.
Solution 2: Pipe cleaners.
Available at any craft store (Michaels) or some dollar stores are pipe cleaners. Originally meant for smokers to clean their pipes, they are more generally useful. Basically a stiff wire with synthetic bristles sticking out in all directions.
In general I am not a fan of non-cleanable surfaces. I had a contigo coffee cup that required rinsing immediately after use (I use both milk and sugar in my coffee) Failing to do this resulted in a bad taste. After boiling the lid in bleach, salt water, stain remover, hydrogen peroxide, I was able to get some blobs of green-black slime out of it, and removed the taste. A month later, despite trying to rinse as soon as I got home each day, the taste was back. I've discarded the cup, and now have a Thermos brand one that can be taken apart when needed.
Solution 3: Different water bottle.
Discard this water bottle and get one that doesn't use a straw.
Ignore it. It's just dust and dirt. You probably get as much dirt on any veg you pull from your garden.