We present a Bayesian method to cross-match 5,827,988 high proper motion Gaia sources ($mu>40 mas yr^{-1}$) to various photometric surveys: 2MASS, AllWISE, GALEX, RAVE, SDSS and Pan-STARRS. To efficiently associate these objects across catalogs, we develop a technique that compares the multidimensional distribution of all sources in the vicinity of each Gaia star to a reference distribution of random field stars obtained by extracting all sources in a region on the sky displaced 2$^prime$. This offset preserves the local field stellar density and magnitude distribution allowing us to characterize the frequency of chance alignments. The resulting catalog with Bayesian probabilities $>$95% has a marginally higher match rate than current internal Gaia DR2 matches for most catalogs. However, a significant improvement is found with Pan-STARRS, where $sim$99.8% of the sample within the Pan-STARRS footprint is recovered, as compared to a low $sim$20.8% in Gaia DR2. Using these results, we train a Gaussian Process Regressor to calibrate two photometric metallicity relationships. For dwarfs of $3500<T_{eff}<5280$ K, we use metallicity values of 4,378 stars from APOGEE and Hejazi et al. (2020) to calibrate the relationship, producing results with a $1sigma$ precision of 0.12 dex and few systematic errors. We then indirectly infer the metallicity of 4,018 stars with $2850<T_{eff}<3500$ K, that are wide companions of primaries whose metallicities are estimated with our first regressor, to produce a relationship with a $1sigma$ precision of 0.21 dex and significant systematic errors. Additional work is needed to better remove unresolved binaries from this sample to reduce these systematic errors.
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