The origins of astrophysical high-energy neutrino flux remain uncertain. Core-collapse supernovae with strong CSM interaction (Type IIn a.k.a. SNe IIn) are compelling candidates for efficient hadronic acceleration and neutrino production. We investigate the possible association between the Type IIn supernova SN2025cbj and the IceCube high-energy neutrino IceCube-250421A, and assess whether the observed properties of the SN permit an appreciable neutrino yield. We combined rapid optical follow-up with LAST and archival ZTF photometry with spectroscopy from LT/SPRAT and MMT/BINOSPEC to characterize the SN's evolution and CSM interaction. We estimated the explosion and peak times from early light-curve fitting, and quantified the chance-coincidence probability with resampling simulations that scramble neutrino right ascensions while preserving declinations and error contours. Using a simple post-shock-breakout interaction model in a dense wind, we estimated the expected muon-neutrino yield for IceCube's real-time Bronze stream. Spectra of SN2025cbj obtained after the neutrino epoch show persistent narrow Balmer lines superposed on broad Lorentzian electron-scattering wings, consistent with sustained dense-CSM interaction. For the multi-messenger association, resampling simulations against the TNS catalog give a chance-coincidence probability for observing k>=1 events of p=~0.24 (and p=~0.078 against the ZTF-BTS catalog). These values are sensitive to the size of the SNe and neutrino samples. A post-breakout interaction scenario predicts an expected N{nu}{mu}~10^-3^ events in the IceCube Bronze alert stream over 76 days per this one candidate. We discuss the implications of these numbers and possible biases that may affect these results.