I suspect that's the trouble. I've been told that heavy callouses on the human hand actually lessen accidental shock; the callous acts as an insulator.
Increased voltage will overcome this natural insulation but rather than spending money on a fence charger, try a conductive gel like vaseline on the wires. Fair warning: if this actually works, you'll probably KILL the bird.
I think you’re right about using a conductive gel. Probably not vaseline though, which I doubt is conductive. There are specially made greases that contain carbon or metal particles that are conductive. The only problem is the good birdies would get their feets greasy.