We study the recently introduced fair division concept of the happy nucleolus for cost allocation among players in a cooperative game, with special focus on its computation. The happy nucleolus applies the same fairness criterion as the well-established nucleolus but with reduced total value. Still, we show that the relation between the two concepts is quite involved, and intuitive properties do not hold - e.g., the entry of a player in the happy nucleolus can be larger than the entry of the same player in the nucleolus, even for monotone and subadditive games. This refutes conjectures of Meir, Rosenschein and Malizia (2011). Further, we study the separation problem of the linear programs appearing in the MPS scheme for computing the (happy) nucleolus. It includes linear subspace avoidance constraints, which can be handled efficiently for problems with a certain dynamic programming formulation due to Köhnemann and Toth (2020). We show how to get rid of these constraints for all monotone games if we allow for an arbitrarily small error of epsilon, thus conserving known approximation guarantees for the same problem without subspace avoidance. Finally, we focus on practical results at the example of vehicle routing games by designing an efficient heuristic based on our previous insights and past work, and demonstrate its power.