According to Bentham and Rawls, should the aim of government be the promotion of happiness?
The aim of government, according to Bentham and Rawls, is utility, defined as "the greatest good for the greatest number." This theory expounds that the goal of the individual is happiness and thus that the aim of humanity is the same. Thus, the aim of government is to provide actions that produce the most pleasure and the least pain. They do this by restricting freedoms as little as possible while maintaining the natural rights of all.
However, both theorists note that utilitarianism isn't perfect. Selfish desires can impede the process. In some cases, it isn't possible to please everyone, or even most people. Situations provide direct contrasts, so Rawls and Benthem, then note that government should act in they manner that is best for most citizens.
In theory, yes, the goal of government, according to Benthem and Rawls is for its citizens to be happy. They attempt to achieve this through the concept of utilitarianism.
In practice, this goal is not realistic. Inherent in utilitarianism is the idea that some minority will be left out, unhappy with that decision.