No, a cap and trade system will not be more fair to energy consumers than a carbon tax, because it will raise prices artificially to all consumers. This is on purpose, so that consumers will decide to use less energy because they can no longer afford it. By design, the cap and trade system will raise prices.
Cap-and-trade systems were established to help companies appear as if they are helping the environment when in reality it does not slow emissions in the first place. These systems are particularly unfair to consumers who must be held to a similar standard and do not have the capital to afford trading.