The 10A rating is not the current draw of the receiver, but the rating of the contacts in the relay, so don't let that put you off.

You'll need a 12V battery (not very big, maybe 500mAh?) and a lamp that works on 12V. I imagine the clever use of PVC pipe and an automotive lamp will do the trick.

The battery capacity you'll need will be about (lamp current)x(hours of operation). This formula ignores the relay receiver current but will give you a rough idea of what mAh battery you'll need.