03 Camry one click, no start, but lights go out on key turn

The battery is too weak to give more than a click of the starter. During that click, the voltage drops so much it makes the lights go out. So if the battery is so new we can assume it's good, then the alternator is not recharging it properly. Fully recharge (don't jump) the battery and drive to the place you bought the battery and have it tested along with the alternator. Then get directly to the repair shop if you need alternator installed, as the battery won't last indefinitely on a recharge.