CURLING -- II. Improvement on the $H_{0}$ Inference from Pixelized Cluster Strong Lens Modeling

Strongly lensed supernovae (glSNe) provide a powerful, independent method to measure the Hubble constant, $H_{0}$, through time delays between their multiple images. The accuracy of this measurement depends critically on both the precision of time delay estimation and the robustness of lens modeling. In many current cluster-scale modeling algorithms, all multiple images used for modeling are simplified as point sources to reduce computational costs. In the first paper of the CURLING program, we demonstrated that such a point-like approximation can introduce significant uncertainties and biases in both magnification reconstruction and cosmological inference. In this study, we explore how such simplifications affect $H_0$ measurements from glSNe. We simulate a lensed supernova at $z=1.95$, lensed by a galaxy cluster at $z=0.336$, assuming time delays are measured from LSST-like light curves. The lens model is constructed using JWST-like imaging data, utilizing both Lenstool and a pixelated method developed in CURLING. Under a fiducial cosmology with $H_0=70\rm \ km \ s^{-1}\ Mpc^{-1}$, the Lenstool model yields $H_0=69.91^{+6.27}_{-5.50}\rm \ km\ s^{-1}\ Mpc^{-1}$, whereas the pixelated framework improves the precision by over an order of magnitude, $H_0=70.39^{+0.82}_{-0.60}\rm \ km \ s^{-1}\ Mpc^{-1}$. Our results indicate that in the next-generation observations (e.g., JWST), uncertainties from lens modeling dominate the error budget for $H_0$ inference, emphasizing the importance of incorporating the extended surface brightness of multiple images to fully leverage the potential of glSNe for cosmology.