Of all challenges arrayed against global 21-cm cosmology experiments aimed at probing the early epochs of the Universe (the Dark Ages and Cosmic Dawn), such as the NASA-funded mission concept DAPPER, the problem of modelling the diffuse galactic beam-weighted foreground at low-frequencies remains one of the most singular, as it is 4-6 orders of magnitude larger than the predicted cosmological signal. In order to characterize and model the beam-weighted foreground, we present an analysis combining analytical and observational models of both the spectral index and galactic sky brightness temperature with simulations of beams having various angular and spectral dependencies. Each combination (spectral index model, sky temperature map, and beam) creates a unique beam-weighted foreground. We generate optimal basis eigenvectors to fit each foreground model using Singular Value Decomposition (SVD), and examine the effects of varying the beam-weighted foreground upon those eigenvectors. We find that the SVD eigenvectors for each unique foreground model are nearly identical when an achromatic, isotropic beam is used—the ideal case. However, when a beam with angular structure and chromaticity weighs the foreground, the resultant SVD eigenvectors are shifted in frequency and amplitude from the ideal case. Furthermore, the beam exposes the intrinsic structural differences in each foreground model, as each model’s SVD eigenvectors are not uniformly shifted in both frequency and amplitude. As such, we conclude that because the beam strongly couples the spatial and spectral structure of the foreground through its chromaticity, in order to extract the 21-cm signal it is essential for the beams to be well-measured and characterized. The power in using our SVD analysis lies in the fact that we do not need to invoke perfect knowledge of the beam or of the foreground in order to separate them from the signal; rather, we need only know all of the possible ways in which they can vary to properly account for such systematic uncertainties. By modelling these variations, we can produce an optimal basis to fit any beam-weighted foreground model, including those with highly chromatic beams.