You should note that variacs are autotransformers, so they don't isolate. You should have an isolation transformer as well.
You don't say what the mains voltage is, a step-down transformer will usually be an isolation transformer anyhow.
There will have to be a capacitor to keep the ripple down. A battery will have no ripple, and I don't know how much some ripple will affect your circuits. If you are taking 1 amp, you need about 10,000 uF to keep the ripple below 1 V. That will be quite an expensive capacitor.