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TeV-scale dark matter is well motivated by notions of naturalness as the new physics threshold is expected to emerge in the TeV regime. We extend the Standard Model by adding an arbitrary SU(2) dark matter multiplet in non-chiral representation. The pseudo-real representations can be viable DM candidates providing that one includes a higher dimensional mass-splitting operator, which avoids the tree-level inelastic scattering through Z-boson exchange. These effective operators give rise to sizable contributions from Higgs mediated dark matter interactions with quarks and gluons. A linear combination of the effective couplings named $λ$ is identified as the critical parameter in determining the magnitude of the cross-section. When $λ$ is smaller than the critical value, the theory behaves similar to the known renormalisable model, and the scattering rate stays below the current experimental reach. Nevertheless, above the criticality, the contribution from the higher dimensional operators significantly changes the phenomenology. The scattering amplitude of pseudo-real models will be coherently enhanced, so that it would be possible for next generation large-exposure experiments to fully probe these multiplets. We studied the parameter space of the theory, taking into account both indirect astrophysical and direct search constraints. It is inferred that multi-TeV mass scale remains a viable region, quite promising for forthcoming dark matter experiments.