Robust a priori error estimates are derived for the unfitted meshe semi-implicit coupling scheme recently introduced in [26], for the simulation of incompresible fluid-structure interaction involving thinwalled solids. The analysis shows that, under a hyperbolic-CFL condition, the leading term in the energy error scales as $O(h^{r−1/2})$, where $r=1, 2$ stands for the extrapolation order of the solid velocity in the viscous luid sub-step. The theoretical findings are illustrated via a numerical experiments which show, in particular, that the considered method avoids the spatial non-uniformity issues of standard loosely coupled schemes and that it delivers practically the same accuracy as the fully implicit scheme.