Powering 12V / 5W device at minimum cost and steady life duration
You need a 5W 12V load supplied for at least 1h or 5Wh

Each “holy grail” LG battery has V=3.6V nom. with a 1C dropout of 3.0V @3Ah or 3.3Vavg*3Ah=10Wh which will support a 2h load with a DC-DC boost converter with 90~95% efficiency from TI WebBench designs or off-the-web purchase.

You will also need a matched charger and undervoltage protection (UVP) on the boost regulator.