Since the effect is eye-candy I recommend starting with a dark plastic card to get the shine and avoid any porosity issues. Then print the interesting gradients with the metallic inks like you mentioned, overlapping with a reasonably registrable dot pattern. I doubt anybody is going to scrutinize it so much that a slight moire will be a deal-breaker. It may add to the psychedelic look.
Then the other side of the card can be printed with white, lighter colors or whatever you see fit, but the black plastic will have that "shiny parking lot" look without having to flood it with varnish.
There is a paper called Currency Paper from CTI Paper that has metallic finishes: silver, copper, and gold. You could look into those papers and then printing your 4-color process over that.
I just printed a metallic silver on ArjoWiggins' Curious Collection. I used the Ice Silver color. Really cool results. You may also want to think about a foil stamp. I'm looking in Crown Roll Leaf Inc.'s catalog and am finding several versions of "oil rainbow" foils. Some can be overprinted, some not.
I love both your suggestions. I remember seeing several interesting special-effects foils in Crown's swatch book several years ago — but I recently looked into stamping iridescent foil and, try as we might, we could not find a source for it (at least not a true iridescent). Crown does offer some holographic foils, which offer true color change effect. The Curious papers are also great if you want a full background.
Depending on your volume, you can get iridescent film laminated to board. If you don't have sufficient volume, you can at least approach a film supplier like Aurora and see if they can recommend a supplier of laminated board that uses their film. They may have some leftover material on their floor that they might be happy to give you, or they may be able to give you off-cuts from a bigger run.
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