[Jared]

I think now that my answer is wrong because I must have a net force in the x direction due to friction. For my answer I had used my previous method of breaking the velocity into two perpendicular components, with one of those components tangent to the circle centered at the robots center and intersecting the wheel.
I then multiplied each of these velocities by the ratio of weight over the wheel to total weight so that the wheel closer to the center of mass and with the most weight on top of it have a greater impact on total velocity.

To me, this seems like you might be leading Us toward forward kinematics for swerve...