Super-sample covariance (SSC) is an important effect for cosmological analyses using the deep structure of the cosmic web; it may, however, be non-trivial to include it practically in a pipeline. Here we lift up this difficulty by presenting a formula for the precision (inverse covariance) matrix and show applications to update likelihood or Fisher forecast pipelines. The formula has several advantages in terms of speed, reliability, stability, and ease of implementation. We present an analytical application to show the formal equivalence between 3 approaches to SSC: (i) at the usual covariance level, (ii) at the likelihood level, and (iii) with a quadratic estimator. We then present an application of this computationally efficient framework to study the impact of inaccurate modeling of the SSC responses for cosmological constraints from stage IV surveys. We find that a weak lensing-only analysis is very sensitive to inaccurate modeling of the scale dependence of the response, which needs to be calibrated at the ~15% level. The sensitivity to this scale dependence is less severe for the joint weak lensing and galaxy clustering analysis (also known as 3x2pt). Nevertheless, we find that both the amplitude and scale-dependence of the responses have to be calibrated at better than 30%.