Don't try and distill H2SO4, it won't work. All that will happen is you getting some SO2 poisoning. What happens is the water will boil off first, because it has a lower boiling point. But when the H2SO4 gets concentrated enough, it will decompose into H2O and SO3, and the SO3 will partially decompose to SO2. That's the problem with H2SO4, it doesn't distill, at least under normal pressure. All you can do is drive off the water it contains, and then it will start to decompose.
And as for battery acid, it does contain PbSO4. Don't use it.