[morzh]

So, I tried various heatshrink tubing pieces, and, even though I think it could work, it is inconvenient. It had to be cut at the top rim of the socket, as otherwise the bulb would not go in. But that exposed the metal with very think insulation (which is what heatshrink is).
So I ordered some 3/8" ID, 1/2" OD silicon tubing, and made this

   

This made fairly thick insulation, close to the original, plus it would not start pulling off the bottom due to some incoming wires' thickness over the socket.

The result is this:

   
   


Plays well, and now ready for aloignment. Hopefully, tomorrow.