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Detecting hidden confounding is crucial for reliable causal analysis from observational data, directly determining which downstream causal inference method to be deployed. Inspired by the theory of higher-order regression, recent sample-efficient hypothesis testing strategies overcome the restrictive requirement of multiple heterogeneous data environment. Despite their progress on single-environment confounder detection, such methods suffer from intrinsic flaws that the structural functions of the causal models should be specified in prior (linear or specific kernel functions). By contrast, real-world data acquisition exhibits diverse, unknown forms of structural functions, imposing an important but challenging gap between theories of higher-order regressions to practical confounder detection. In this paper, we contribute a Bi-level Kernel Confounder Detection (BiKCD) framework by learning adaptive kernelized structural space of structural functions. Subsequently, our BiKCD constructs hypothesis testing by comparing coefficients from the higher-order regression and the classical ordinary least squares in learned kernelized space. Finally, the hypothesis is calibrated to ensure valid inference under adaptivity. Theoretically, we establish an oracle-type risk bound for the selected structural space over a candidate kernel family, with the Type-I error control for the downstream test. Extensive experiments on synthetic and real-world datasets demonstrate the effectiveness of the proposed BiKCD.