I think the feeling that you are mentioning could be more appropriately called being at peace, purposefulness, meaningfulness, mindfulness etc. It is quite different from the western concept of happiness. I don't think Gandhi or his disciples were singing and dancing when cleaning the toilets. They approached this duty solemnly. This approach to life is indicated in the Gita where it says that you need do things because they are your Karma without any expectation of reward(read happiness) in return. In this view of life, happiness and sadness are both transient and therefore it is useless to pursue happiness or run away from sadness. This perspective is thus tangential to the western pursuit of happiness. In fact, I think it can be argued that the pursuit of happiness is the real cause of sadness.

Apart from Karma-based approach in Gita, you can also look towards the Service-based approach in Sikhism or the Non-Detachment in Buddhism. In Sikhism, service is an integral part of religion and every Sikh is expected to give at least some time to do voluntary service in Gurudwaras(Sikh Temples). This is mostly in the form of chores like cooking food, washing utensils, polishing the shoes of visitors etc. In India, you can find some of the richest Sikhs doing these chores in Gurudwaras.

Similarly in Buddhism, detachment of the self is an important aspect of living a life which involves restraining oneself, cultivating discipline and practicing mindfulness and meditation.

None of them have any details about pursuing happiness specifically.