You could try holding the twisted end of each joint in the flame of a gas cigarette lighter for a few seconds to burn off any insulation.
It's a technique sometimes used on some thin wires & especially "tinsel cable".
As you have already twisted the wires, put around 6 - 10mm of the twisted wire in to the
blue part of the gas flame; don't put it over the top of the flame.
Then untwist and re-twist it to get any traces of burnt plastic or lacquer out of the joint. It should look bright rather than black.
You could also fold the twisted section in half to make it more compact before taping it.
It may be worth practicing on some bits of the wire you cut out, to judge how long it needs to get rid of the insulation fully, without damaging the copper strands.
If there is enough room, you could add some very small nylon "chocolate block" screw terminals in individual sections to ensure the joints do not fail over time, putting the twisted wire joint in to just one side of each section and tightening the screw gently to hold the twisted connection solidly.
This style stuff:
Durite 15 Amps nylon cable connector strip at Arc Components Ltd. Strip of 12 blocks for automotive cable and vehicle wiring, with steel screw and brass insert.
www.arc-components.com