Very thin, flexible wire for LEDs -> Arduino inside cramped space?
There are all kinds of wire.  Maybe for your case "wire wrap" wire is applicable since it's available and cheap.  This is usually quite thin, like 30 guage.  Of course wire this thin isn't too robust.  If you're sure you only need to flex it a few times and then it's fixed in place, wire wrap wire might do it.

I think you're going to be fiddling with this thing more than you realize.  Wire wrap wire, or any single-strand wire, will break after enough flexes.  Is the wire size really that big a deal?  Of course 14 gauge house wiring would be silly, but it's hard to imagine ordinary stranded 24 guage hookup wire being big enough to be a problem.  You may want to get some with relatively low insulation voltage so that the insulation will be smaller in diameter.