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The Legacy Survey of Space and Time (LSST) with the Vera Rubin Observatory will provide strong microlensing constraints on dark compact objects (DCOs) in our Galaxy. However, most current forecasts limit their analysis to Primordial Black Holes (PBH). It is unclear how well LSST microlensing will be able to constrain alternative models of DCOs with different Galactic spatial profile distributions at a subdominant DM fraction. In this work, we investigate how well LSST microlensing will constrain spherical or disk-like Galactic spatial distributions of DCOs, taking into account extended observing times, baryonic microlensing background, and sky distribution of LSST sources. These extensions represent significant improvements over existing microlensing forecasts in terms of both accuracy and versatility. We demonstrate this power by deriving new LSST sensitivity projections for DCOs in spherical and disk-like distributions. We forecast that LSST will be able to constrain one solar mass PBHs to have a DM fraction under $4.1\times10^{-4}$. One-solar-mass objects in a dark disk distribution with the same dimensions as the Galactic disk will be constrained below $3.1\times10^{-4}$, while those with $m = 10^5M_{\odot}$ will be constrained to below $3.4\times10^{-5}$. We find that compressed dark disks can be constrained up to a factor of $\sim10$ better than ones with identical dimensions to the baryonic disk. We also find that dark disks become less tightly constrained when they are tilted with respect to our own disk. This forecasting software is a versatile tool, capable of constraining any model of DCOs in the Milky Way with microlensing, and is made publically available at {https://github.com/HarrisonWinch96/DarkDisk_Microlensing}.