Circular $e^{+}e^{-}$ colliders operating at energies below the di-Higgs production threshold can provide information on the trilinear Higgs self-coupling $\lambda_{hhh}$ via its loop contributions to single Higgs production processes and electroweak precision observables. We investigate how well a non-SM value of $\lambda_{hhh}$ can be determined indirectly via its loop contributions within a global EFT fit. Using an inert doublet extension of the SM Higgs sector as an example for a scenario of physics beyond the SM that could be realised in nature, we find that theoretical uncertainties related to the treatment of loop contributions and the truncation of the EFT expansion, which are usually neglected, play an important role in determining the sensitivity to $\lambda_{hhh}$ in a global fit. The results obtained from such an indirect determination of $\lambda_{hhh}$ without taking these additional uncertainties into account would be too optimistic, leading to an artificially high resulting precision for $\lambda_{hhh}$. They could therefore be misleading in the quest to precisely identify the underlying physics of electroweak symmetry breaking.