After reviewing some of the fundamental aspects of Drinfel'd doubles and Poisson-Lie T-duality, we describe the three-dimensional isotropic rigid rotator on $SL(2,\mathbb{C})$ starting from a non-Abelian deformation of the natural carrier space of its Hamiltonian description on $T^*SU(2) \simeq SU(2) \ltimes \mathbb{R}^3$. A new model is then introduced on the dual group $SB(2,\mathbb{C})$, within the Drinfel'd double description of $SL(2,\mathbb{C})=SU(2) \bowtie SB(2,\mathbb{C})$. The two models are analyzed from the Poisson-Lie duality point of view, and a doubled generalized action is built with $TSL(2,\mathbb{C})$ as carrier space. The aim is to explore within a simple case the relations between Poisson-Lie symmetry, Doubled Geometry and Generalized Geometry. In fact, all the mentioned structures are discussed, such as a Poisson realization of the $C$-brackets for the generalized bundle $T \oplus T^*$ over $SU(2)$ from the Poisson algebra of the generalized model. The two dual models exhibit many features of Poisson-Lie duals and from the generalized action both of them can be respectively recovered by gauging one of its symmetries, as it is customary in the framework of the Double Field Theory.