I don’t recall seeing the Lamp Socket cable sold separately, but it seems like a good idea. That made me curious, and then I saw a potentially relevant Reddit post. It’s a few years old, so I don’t know how current any of the product recommendations are. There are also some product recommendations in an older Forum post that might be relevant, but I haven’t vetted any of those, either.
Another thought I had was trying electronics cleaner. You could disconnect the cable from the Cam v3 and the Lamp Socket and unscrew the Lamp Socket from its E26 socket and then try electronics cleaner on the connections before putting it all back together and trying again. I don’t know anything about where you’ve been using the socket and camera, but I wonder if you’ve gotten some oxidation or corrosion on the contacts and if electronics cleaner would restore the connections for you. That’s something else I tried while troubleshooting this problematic Cam OG, and the thing that would pretty consistently get the camera powered up again (at least for a short time) was blowing on/into the Micro-B connector of the OEM cable righ before plugging it back into the Cam OG. (Yes, blowing with my mouth, just like on an old home video game system cartridge.) I don’t know what that was actually doing to the connector itself, but as dumb as it probably sounds that helped me to narrow the problem to that part of the system, and replacing the cable has solved that problem for me.