Optimizing structural designs, especially for complex systems like turbine blade cooling structures, requires efficient strategies for handling categorical configurations alongside computationally expensive simulations. This article presents a Bayesian optimization strategy tailored for integrated tasks involving categorical configurations and high-dimensional continuous design variables. A Gaussian process with Con-Cat kernel is proposed in order to merge response datasets from diverse configurations effectively, capturing inter-configuration and intra-configuration correlations seamlessly. Additionally, a supervised dimension–reduction scheme is developed based on subspace activation, utilizing a half-Cauchy distribution. Remarkably, the Con-Cat kernel represents a generalization of standard kernels, achieving equivalence in scenarios solely involving continuous variables. The subspace activation scheme enhances surrogate modelling performance without introducing extra model parameters, which is particularly beneficial for sparse datasets. Numerical evaluations, including three mathematical functions, a supported beam problem, and turbine blade cooling structure design optimization, demonstrate the superiority of the proposed Bayesian optimization strategy, exhibiting up to an 85% improvement over five alternative approaches, especially in scenarios with sparse data.