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I'm afraid not. A Triple Triad card is simply a species number, with no other way of differentiation. Changing this would require a non-trivial change to the code.
You could try creating a new species just for the shiny card. Make sure this new species isn't used elsewhere, so it won't appear in any Pokédex and so on.
You'd have to change some code. The existing card shop figures out which cards it should stock somehow, so copying it all and changing that part would be a good start.
The Triple Triad minigame is quite bare-bones and doesn't have a lot of things it should (such as a way to view your cards outside of a duel, or deck-building). I'd quite like to overhaul it, but I have more pressing matters to attend to first.