In this work, we study a three-parameter family of Bell functionals in a bipartite scenario with 3 measurement settings per party and 2 outcomes per measurement. The members of this family can be thought of as variations of the well-known I3322 functional, the only one in this scenario corresponding to a tight Bell inequality if we take aside the CHSH inequality. An analysis of their largest value achievable by quantum realisations (quantum value) naturally splits the set into two branches, and for the first of them, we show that this value is given by a simple function of the parameters defining the functionals. In this case we completely characterise the realisations attaining the optimal value and show that these functionals can be used to self-test any partially entangled state of two qubits. The optimal measurements, however, are not unique and form a one-parameter family of qubit measurements. Within the second branch, the quantum value presents a more complex dependence on the parameters defining the functionals and is studied numerically, identifying first the regions in parameter space where two-qubit systems suffice to approach the quantum value. The remainder of the branch includes the I3322 functional, for which a particular sequence of finite-dimensional realisations introduced by K. Pál and T. Vértesi is known to numerically attain the quantum value in the limit of infinite local dimension. We study the performance of these special strategies beyond the I3322 case and analyse the optimal solutions in those cases where they succeed in approaching the quantum value.