When you move around an unanchored part with your character and the part moves, the part movement replicates because network ownership is transferred over to the client. But when a part is anchored/welded physics aren’t simulated on it so it can’t move so there is no point in making the client the network owner.
If anyone is curious, I found a solution that works for me. I just added a BodyVelocity and set BodyVelocity.Velocity to 0, 0, 0 and set BodyVelocity.MaxForce to math.huge, math.huge, math.huge. I did something similar with a BodyGyro. Allow my character to move I created a custom character movement script that updated the BodyVelocity.Velocity as needed.