Dual-phase titanium alloys, used in aircraft engines and structures, can develop strong crystallographic texture during thermo-mechanical processing, which affects the fatigue life and makes the material anisotropic. During hot working, these alloys undergo simultaneous co-deformation of HCP $\alpha$ and BCC $\beta$ phases, phase transformation and recrystallization, all of which can contribute to texture development. These mechanisms are well understood in isolation, however they interact in complex ways during processing making it challenging to fully unravel their relative importance towards texture development and how it changes with processing conditions like temperature and strain rate. In this work, a full-field crystal plasticity model is used to model the contribution of plastic deformation to the texture development during hot uniaxial compression of a sample containing a large ($\SI{1}{\cm}$), single-orientation HCP $\alpha$ colony, grown from a single prior BCC $\beta$ grain. The relative slip resistances used in the model were calibrated using data from in-situ synchrotron experiments. The results of these simulations are compared to experimental electron backscatter diffraction (EBSD) results of single colony hot compression tests. The simulations showed that although the beta is the much softer phase, plastic deformation extends to the alpha phase due to stress concentrations at micro-structural features. This makes deformation in the single colony highly heterogeneous, which generally leads to texture weakening in both phases. The results show  good general agreement with those of the experiments, however new orientations are found in the experiments that cannot be predicted using crystal plasticity modelling. These results suggests texture development by crystallographic slip cannot, on its own, explain the texture development in these materials and that future models of texture prediction should include the effects of recrystallizaton.