Battery power in the home: Fire Safety
You will almost certainly want rechargeable batteries for this, but AA cells will only give you 2-3 hours use from fully charged (usual capacity is 1.5-2.7Ah).

12V rechargeable battery packs are readily available for emergency lighting etc, so that part shouldn't be hard to deal with.

For safety, you should choose protection which will break the circuit before any components (LED tape, wires etc) melt or get so hot that something next to them burns.  If you are sure your normal current will never exceed 1A, then you might choose a 1.5A fuse, and ensure your wires are big enough to carry 5-6A without overheating.