Galactic white dwarf binaries (WDBs) and black hole binaries (BHBs) will be gravitational wave (GW) sources for LISA. Their detection will provide insights into binary evolution and the evolution of our Galaxy through cosmic history. Here, we make predictions of the expected WDB and BHB population within our Galaxy. We combine predictions of the compact remnant binary populations expected by stellar evolution by using the detailed Binary Population and Spectral Synthesis code (BPASS) with a Milky Way analogue galaxy model from the Feedback In Realistic Environment (FIRE) simulations. We use \textsc{PhenomA} and \textsc{LEGWORK} to simulate LISA observations. Both packages make similar predictions that on average four Galactic BHBs and 673 Galactic WDBs above the signal-to-noise ratio (SNR) threshold of 7 after a four-year mission. We compare these predictions to earlier results using the Binary Star Evolution (BSE) code with the same FIRE model galaxy. We find that BPASS predicts a few more LISA observable Galactic BHBs and a twentieth of the Galactic WDBs. The differences are due to the different physical assumptions that have gone into the binary evolution calculations. These results indicate that the expected population of compact binaries that LISA will detect depends very sensitively on the binary population synthesis models used and thus observations of the LISA population will provide tight constraints on our modelling of binary stars. Finally, from our synthetic populations we have created mock LISA signals that can be used to test and refine data processing methods of the eventual LISA observations.