Within its observational band the Laser Interferometer Space Antenna, LISA, will simultaneously observe orbital modulated waveforms from Galactic white dwarf binaries, a binary black hole produced gravitational-wave background, and potentially a cosmologically created stochastic gravitational-wave background (SGWB). The overwhelming majority of stars end their lives as white dwarfs, making them very numerous in the Milky Way. We simulate Galactic white dwarf binary gravitational-wave emission based on distributions from various mock catalogues and determine a complex waveform from the Galactic foreground with 3.5 × 10^7 binaries. We describe the effects from the Galactic binary distribution population across mass, position within the Galaxy, core type, and orbital frequency distribution. We generate the modulated Galactic white dwarf signal detected by LISA due to its orbital motion, and present a data analysis strategy to address it. The Fisher Information and Markov Chain Monte Carlo methods give an estimation of the LISA noise and the parameters for the different signal classes. We estimate the detectable limits for the future LISA observation of the SGWB in the spectral domain with the three LISA channels A, E, and T. We simultaneously estimate the Galactic foreground, the astrophysical and cosmological backgrounds. Assuming the expected astrophysical background and a Galactic foreground, a cosmological background energy density of around Ω_GW,cosmo ≈ 8 × 10^−13 could be detected by LISA. LISA will either detect a cosmologically produced SGWB, or set a limit that will have important consequences.