Yes, you can just hang the water bottle(s) with the food.

Rodents can climb down the line to the hanging bag, but they don't all do this. I saw a woodrat do it; he/she had climbed down from the tree on the parachute cord. But it doesn't seem to be terribly common, at least in areas where they haven't yet learned this trick. Once they have learned something like this, and it works for them, they are often quick to utilize the same approach again when the opportunity presents itself.

There may be ways to prevent this. A lightweight plastic collar might be one of them. There are probably details on how to construct an effective rodent collar somewhere online. Maybe one source of information would be websites that deal with protecting hanging bird feeders. Wild Birds Unlimited might have some information. A very light weight cone-shaped collar could be made with tabs that would allow it to be flattened for transport.

Those peanut jars are light weight, and can be filled with food and other items, so they don't take up much space. They can also be used for extra water if needed.