Absolutely! I think that's what the author meant when he said "In my prototype, your hand at the bottom of the screen is a background (similarly, in my E-Reader solitaire game, all of the cards are drawn into a background instead of using sprites)."
One way to handle that is to provide all the possible tile variations, but that would take up so much space. So you'd have a set of tiles for a regular Ace of Spades, and an entirely different group of tiles for a Lucky Ace of Spades for example.
The GBA has 4 backgrounds, so it would be doable to grab three of them and use them to render cards. That would only leave 1 background left for, well, the background :)
Another option would be to use a memory buffer and implement tile rendering yourself that accounts for transparent pixels. That would be the best of backgrounds and sprites combined into one. That would solve many problems, at the cost of the implementation would probably take up a lot of space. My hunch is this would be the best approach.
This right here might be why I find this platform so interesting. It's very limited, and the limitations usually bump into each other and you often steel from Peter to pay Paul.
Oh and the post didn't mention debuffed cards (they have a red X drawn over them). That'd be yet another card layer to throw into this mix.