The emission and escape of Lyman-α photons from star-forming galaxies is determined through complex interactions between the emitted photons and a galaxy's interstellar and circumgalactic gas. This causes Lyman-α emitters (LAEs) to commonly appear not as point sources but in spatially extended halos with complex spectral profiles. We developed a 3D spatial-spectral model of Lyman-α halos (LAHs) to replicate LAH observations in integral field spectroscopic studies, such as those made with VLT/MUSE. The profile of this model is a function of six key halo properties: the halo- and compact-source exponential scale lengths (r_sH and r_sC), the halo flux fraction (f_H), the compact component ellipticity (q), the spectral line width (σ), and the spectral line skewness parameter (γ). Placing a series of model LAHs into datacubes that reflect observing conditions in the MUSE UDF-Mosaic survey, we tested their detection recoverability and determine that σ, r_sH, and f_H are expected to have the most significant effect on the detectability of the overall LAH at a given central wavelength and intrinsic line luminosity. We developed a general selection function model that spans a grid of these halo parameters. Using it with a sample of 145 LAHs with measured halo properties observed in the UDF-Mosaic survey, we derived completeness-corrected, intrinsic distributions of the values of σ, r_sH, and f_H for 3 < z < 5 LAHs. We present the best-fit functional forms of the distributions as well as a σ distribution corrected for instrumental line-spread function broadening, and thereby show the physical line-spread distribution of the intrinsic population. Finally, we discuss possible implications for these distributions for the nature of Lyα emission through the circumgalactic medium, finding that observations may undercount LAHs with extended halo scale lengths compared to the intrinsic population.