We discuss asymptotics of large Boltzmann random planar maps such that every vertex of degree $k$ has weight of order $k^{-2}$. Infinite maps of that kind were studied by Budd, Curien and Marzouk. These maps can be seen as the dual of the discrete $\alpha$-stable maps studied by Le Gall and Miermont for $\alpha=3/2$ or as the gaskets of critical $O(2)$-decorated random planar maps. We compute the asymptotics of the graph distance and of the first passage percolation distance between two uniform vertices, which are respectively equivalent in probability to $(\log \ell)^2/\pi^2$ and $2(\log \ell)/(\pi^2 p_{\bf q})$ when the perimeter of the map $\ell$ goes to $\infty$, where $p_{\bf q}$ is a constant which depends on the model. We also show that the diameter is of the same order as those distances for both metrics and obtain in particular that these maps do not satisfy scaling limits in the sense of Gromov-Prokhorov or Gromov-Hausdorff for lack of tightness. To study the peeling exploration of these maps, we prove local limit and scaling limit theorems for a class of random walks with heavy tails conditioned to remain positive until they die at $-\ell$ towards processes that we call stable L\'evy processes conditioned to stay positive until they jump and die at $-1$.