While the Higgs characterisation programme is well under way, direct signs for new physics beyond the Standard Model remain elusive. Performing a fit of fully differential Higgs production cross sections at the LHC to a subset of Higgs-relevant effective operators, we discuss the extent to which theoretical uncertainties can limit the sensitivity in such a new physics search programme. Extending the dimension-6 Higgs Effective Field Theory framework by introducing new light degrees of freedom that can contribute to an invisible (or undetectable) Higgs decay width h -> phi phi, we show how differential coupling fits can disentangle effects from non-Standard Model couplings and an invisible decay width, as present in many new physics scenarios, such as Higgs-portal dark matter. Including the so-called off-shell measurement that has been advocated as a sensitive determination of the Higgs width in the k framework, we show explicitly that this method does not provide complementary sensitivity for scale-separated new physics A >> m(h) >> m(phi), which is favoured in beyond the Standard Model scenarios that relate astrophysics and collider phenomenology in the light of non-observation of new physics during run 1 of the LHC.