Just as bright as fluorescent and runs off of batteries is very, very unlikely. However, without knowing what fluorescent plug in type you have I would not know. My experience with battery operated lights is they are very dim, or you have to replace the batteries a lot.

How 'bout lights that run on 12vac? Just as bright with few wiring restrictions.

Yes, 12V LED lights will be the most energy-efficient option, and you can get rechargeable batteries for them to, if you don't want to plug them into the wall.