Electrical tape is like duct tape, very handy stuff, but never used to tape electrical joints. It doesn't like heat, solvents or just plain age. It will come loose and fall off, count on it.
So how to insulate a splice? Use insulated crimp-on splices and proper crimping pliers. Count on spending close to $100 for good pliers.
Do not use a soldering iron because the solder will travel up a stranded wire under the insulation.
Don't use wire nuts, they are designed for solid core wire and they don't like vibration and tend to fall off.