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Standing fast sausage modes in flare loops were suggested to account for a considerable number of quasi-periodic pulsations (QPPs) in the light curves of solar flares. This study continues our investigation into the possibility of inverting the measured periods $P$ and damping times $\tau$ of sausage modes to deduce the transverse Alfvén time $R/v_{\mathrm{Ai}}$, density contrast $\rho_{\mathrm{i}}/\rho_{\mathrm{e}}$, and the steepness of the density distribution transverse to flare loops. A generic dispersion relation governing linear sausage modes is derived for pressureless cylinders where density inhomogeneity of arbitrary form takes place within the cylinder. We show that in general the inversion problem is under-determined for QPP events where only a single sausage mode exists, whether the measurements are spatially resolved or unresolved. While $R/v_{\mathrm{Ai}}$ can be inferred to some extent, the range of possible steepness parameters may be too broad to be useful. However, for spatially resolved measurements where an additional mode is present, it is possible to deduce self-consistently $\rho_{\mathrm{i}}/\rho_{\mathrm{e}}$, the profile steepness, and the internal Alfvén speed $v_{\mathrm{Ai}}$. We show that at least for a recent QPP event that involves a fundamental kink mode in addition to a sausage one, flare loop parameters are well constrained even if the specific form of the transverse density distribution remains unknown. We conclude that spatially resolved, multi-mode QPP measurements need to be pursued to infer flare loop parameters. [ABSTRACT FROM AUTHOR]