We describe our approach to solving the problem of ensuring the solenoidality of the magnetic field vector in 3D inversions, as well as the estimation of the uncertainty in the inferred magnetic field. The solenoidality of the magnetic field vector is often disregarded in the inversion of spectropolarimetric data due to limitations in the traditional 1D inversion techniques. We propose a method for ensuring the solenoidal condition in 3D inversions based on our mesh-free approach. The increase in dimensionality with respect to the 1D inversion techniques is such that some of the traditional methods for determining the uncertainties become unfeasible. We propose a method based on a Monte Carlo approach for determining the uncertainty of the magnetic field inference. Due to the physics of the problem, we can compute the uncertainty while only increasing the total required computational time by a factor of about 2. We also propose a metric for quantifying the uncertainty that thus describes the degree of confidence of the magnetic field inference. Finally, we perform a numerical experiment to demonstrate the feasibility of both the method and the metric proposed to quantify the uncertainty.