We study conditional average treatment effect (CATE) estimation from observational data in regimes with potentially weak overlap, when the nuisance response functions lie in a reproducing kernel Hilbert space (RKHS), but the contrast function is structurally simpler. We consider three forms of such simplicity: (i) membership in a lower-complexity RKHS with faster eigenvalue decay, (ii) a source condition relative to the nuisance kernel, or (iii) dependence on a known low-dimensional covariate representation. We develop a unified two-stage kernel ridge regression procedure that remains valid under weak overlap and attains minimax rates determined by the complexity of the contrast function. The procedure avoids propensity-score estimation and separate nuisance convergence-rate assumptions. We also show that a simple model-selection step over candidate contrast spaces and regularization levels satisfies an oracle inequality, enabling adaptation to unknown CATE regularity.

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