Remember to ground the neck plate while you're at it.
Shielding the bridge plate is total overkill. You might find a more robust contact between the bridge and the ground wire if you solder the wire to a section of copper foil under the bridge. Shielding the control plate, if it is metal, is also totally redundant and unnecessary.
OP's bass doesn't have a ground issue. It is noisy until he touches the strings, which means that the ground wire running to the bridge is intact. The detail about touching the strings near the headstock specifically is just additional fluff.
The human body is a good wet, fatty antenna for AC electromagnetic fields. When you take your hands off the strings of an unshielded guitar, the noise is
you. The bridge ground wire diverts this signal to ground, shutting
you up. Shielding mitigates this issue, but isn't the panacea that some claim. I've found that a shielded instrument is quieter regardless of whether my hands are on the strings or not. I achieved the best results with a MIJ Mustang bass.