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Faced with problems of increasing complexity, recent research in Bayesian Optimisation (BO) has focused on adapting deep probabilistic models as flexible alternatives to Gaussian Processes (GPs). In a similar vein, this paper investigates the feasibility of employing state-of-the-art probabilistic transformers in BO. Upon further investigation, we observe two drawbacks stemming from their training procedure and loss definition, hindering their direct deployment as proxies in black-box optimisation. First, we notice that these models are trained on uniformly distributed inputs, which impairs predictive accuracy on non-uniform data - a setting arising from any typical BO loop due to exploration-exploitation trade-offs. Second, we realise that training losses (e.g., cross-entropy) only asymptotically guarantee accurate posterior approximations, i.e., after arriving at the global optimum, which generally cannot be ensured. At the stationary points of the loss function, however, we observe a degradation in predictive performance especially in exploratory regions of the input space. To tackle these shortcomings we introduce two components: 1) a BO-tailored training prior supporting non-uniformly distributed points, and 2) a novel approximate posterior regulariser trading-off accuracy and input sensitivity to filter favourable stationary points for improved predictive performance. In a large panel of experiments, we demonstrate, for the first time, that one transformer pre-trained on data sampled from random GP priors produces competitive results on 16 benchmark black-boxes compared to GP-based BO. Since our model is only pre-trained once and used in all tasks without any retraining and/or fine-tuning, we report an order of magnitude time-reduction, while matching and sometimes outperforming GPs.