We present a comprehensive orbital analysis to the exoplanets $\beta$ Pictoris b and c that resolves previously reported tensions between the dynamical and evolutionary mass constraints on $\beta$ Pic b. We use the MCMC orbit code orvara to fit fifteen years of radial velocities and relative astrometry (including recent GRAVITY measurements), absolute astrometry from Hipparcos and Gaia, and a single relative radial velocity measurement between $\beta$ Pic A and b. We measure model-independent masses of $9.3^{+2.6}_{-2.5}\, M_{\rm Jup}$ for $\beta$ Pic b and $8.3\pm 1.0\,M_{\rm Jup}$ for $\beta$ Pic c. These masses are robust to modest changes to the input data selection. We find a well-constrained eccentricity of $0.119 \pm 0.008$ for $\beta$ Pic b, and an eccentricity of $0.21^{+0.16}_{-0.09}$ for $\beta$ Pic c, with the two orbital planes aligned to within $\sim$0.5$^\circ$. Both planets' masses are within $\sim$1$\sigma$ of the predictions of hot-start evolutionary models and exclude cold starts. We validate our approach on $N$-body synthetic data integrated using REBOUND. We show that orvara can account for three-body effects in the $\beta$ Pic system down to a level $\sim$5 times smaller than the GRAVITY uncertainties. Systematics in the masses and orbital parameters from orvara's approximate treatment of multiplanet orbits are a factor of $\sim$5 smaller than the uncertainties we derive here. Future GRAVITY observations will improve the constraints on $\beta$ Pic c's mass and (especially) eccentricity, but improved constraints on the mass of $\beta$ Pic b will likely require years of additional RV monitoring and improved precision from future Gaia data releases.