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The main goal of this work, the second in a three-paper series, is to testthe impact of a methodological improvement in measuring the magnification biassignal on a sample of submillimeter galaxies and its implications forconstraining cosmological parameters. The analysis considers the angularcross-correlation function between a foreground sample of GAMA galaxies($0.2<z<0.8$) and a background sample of H-ATLAS submillimeter galaxies($1.2<z<4.0$). A refined methodology, discussed extensively in Paper I, isused. By interpreting the weak lensing signal within the halo model andemploying an MCMC algorithm, the posterior distribution of the halo occupationdistribution (HOD) and cosmological parameters is obtained for a flat$\Lambda$CDM model. The analysis incorporates the foreground angularauto-correlation function to account for galaxy clustering. The resultsdemonstrate a remarkable improvement in uncertainties for both HOD andcosmological parameters compared to previous studies. However, when using thecross-correlation data alone, the estimation of $\sigma_8$ depends on priorknowledge of $\beta$, the logarithmic slope of the background number counts.Assuming a physically motivated prior distribution for $\beta$, mean values of$\Omega_m=0.18^{+0.03}{-0.03}$ and $\sigma_8=1.04^{+0.11}{-0.07}$ are obtained.These results may however be subject to an inherent bias in the data due toanomalous behavior observed in the G15 field. After excluding the G15 region,the mean values shift to $\Omega_m=0.30^{+0.05}{-0.06}$ and$\sigma_8=0.92^{+0.07}{-0.07}$. This becomes more apparent when adding theclustering of the foreground sample, but the dependence on $\beta$ informationdisappears, mitigating the aforementioned issue. Excluding the G15 region, thejoint analysis yields mean values of $\Omega_m=0.36^{+0.03}{-0.07}$,$\sigma_8=0.90^{+0.03}{-0.03}$, and $h=0.76^{+0.14}_{-0.14}$.

Methodological refinement of the submillimeter galaxy magnification bias. Paper iI: cosmological analysis with a single redshift bin