You are going to have to find a different pump, because those windscreen washers cannot hold pressure. You need to find a pump with a high enough pressure capability to atomize the mix.
But for now you can use the system you mentioned and later if you want to improve it you can do so. Also if you don't have methanol you can use just water.
Battery power wont be to much of a problem those small battery packs used in those fast rc cars would probably last two days.
The most difficult part would be to make a automatic switch of sorts.
What bike is it?