We introduce and study a new random permutation model that generalizes the $k$-card minimum model defined by Travers and the Mallows model. We calculate the permuton limit of such a sequence of random permutations. As a corollary, we deduce the law of large numbers for pattern densities. Moreover, we prove a universality result about the band structure of the limiting permuton, confirming a conjecture of Travers about the $k$-card minimum model. More specifically, we show that if a certain model parameter goes to infinity then the appropriately scaled restriction of the permuton measure to a line that intersects the diagonal perpendicularly converges weakly to the logistic distribution.