How should I design a power bus for a wearable project?
Drastic edit of my answer, as I didn't fully understand the LED strips.

I would think the best option is very wide, tinned copper ground strap.  They have these with very flexibly braid, used for grounding hinged interfaces.  You should be able to solder individual strip power wires to them and just have to keep them isolated vertically.  Tying the battery at the center of the strip would minimize the voltage difference among all strands.