Your drawing has so narrow strokes that they are difficult to make seem metallic in small size. If only the black strokes should be metallic, then I think you already have as good answer as possible (=the strokes are used to mask a foil image and the result is placed on a black background for good contrast)
Here's another way: Have the white areas as plain foil, present your shape as a fake embossing.
The large white areas and black areas are both separated to different layers and colorized with yellowish gradients. Before making the copies the contrast was increased.
Both have got differently set Bevel&Emboss layer styles. The originally white areas are in the bottom and have wide bevel, the originally black strokes are on top and have got less wide pillow emboss.
The curves layer is for controlling the gloss.
NOTE: Photoshop's Bevel&Emboss has also a customizable complex bevel profile. It's well possible that one Bevel&Emboss in the stroke-based top layer is enough with it.
I saved a copy of the PSD here for inspections:
I guess the result would be better if the white areas were colored with a foil texture photo instead of a gradient. This is a test of it:
It's not especially good. The foil photo is too blurred. Unfortunately I haven't a sharp one.
About GIMP: It hasn't as standard rich enough bevel & emboss effects. Something which at least resembles the results in Photoshop can be found in downloadable extras. I have met it in a Script-Fu collection for the previous version. Searching one for current GIMP is beyond the scope of this answer.