We present an integrated (or end-to-end) framework for the Real2Sim2Real problem of manipulating deformable linear objects (DLOs) based on visual perception. Working with a parameterised set of DLOs, we use likelihood-free inference (LFI) to compute the posterior distributions for the physical parameters using which we can approximately simulate the behaviour of each specific DLO. We use these posteriors for domain randomisation while training, in simulation, object-specific visuomotor policies (i.e. assuming only visual and proprioceptive sensory) for a DLO reaching task, using model-free reinforcement learning. We demonstrate the utility of this approach by deploying sim-trained DLO manipulation policies in the real world in a zero-shot manner, i.e. without any further fine-tuning. In this context, we evaluate the capacity of a prominent LFI method to perform fine classification over the parametric set of DLOs, using only visual and proprioceptive data obtained in a dynamic manipulation trajectory. We then study the implications of the resulting domain distributions in sim-based policy learning and real-world performance.