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We use observations of 66 reverberation-measured \Mgii\ quasars (QSOs) in the redshift range $0.36 \leq z \leq 1.686$ -- a subset of the 78 QSOs we previously studied that also have \rfe\ (flux ratio parameter of UV \Feii\ to \Mgii\ that is used as an accretion-rate proxy) measurements -- to simultaneously constrain cosmological model parameters and QSO 2-parameter and 3-parameter radius-luminosity ($R-L$) relation parameters in six different cosmological models. We find that these QSO $R-L$ relation parameters are independent of the assumed cosmological model and so these QSOs are standardizable through the $R-L$ relations. Also: (1) With the 2-parameter $R-L$ relation, we find that the low-\rfe\ and high-\rfe\ data subsets obey the same $R-L$ relation within the error bars. (2) Extending the 2-parameter $R-L$ relation to a 3-parameter one does not result in the hoped-for reduction in the intrinsic dispersion of the $R-L$ relation. (3) Neither of the 3-parameter $R-L$ relations provide a significantly better fit to the measurements than does the 2-parameter $R-L$ relation. These are promising results for the on-going development of \Mgii\ cosmological probes. The first and third of these results differ significantly from those we found elsewhere from analyses of reverberation-measured \hb\ QSOs.