We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each $α\in (1,2)$ that if the face degree is in the domain of attraction of an $α$-stable Lévy process, the corresponding random planar map has an infinite volume limit in the Benjamini-Schramm topology. We also show in the limit that the properly rescaled contour functions associated with the northwest and southeast trees converge in law to a certain correlated pair of $α$-stable Lévy processes. Combined with other work, this allows us to identify the scaling limit of the planar map with an SLE$_κ(ρ)$ process with $ρ= κ-4 < -2$ on $\sqrtκ$-Liouville quantum gravity for $κ\in (4/3,2)$ where $α, κ$ are related by $α= 4/κ-1$.