By all means post your photo.
My H and I honeymooned in Hawaii 24 years ago and there were "wish pearl" kiosks even then. They target tourists, of course.
Cultured freshwater pearls are anything but rare.
They are never naturally black. If they are black, they are dyed. (Same goes for cultured akoya pearls-- if they are black, then they are dyed.)
Cultured Tahitian pearls
do come by their
color naturally, but the black-lipped mollusks (
Pinctada Margaritifera) they are
cultured in are large, dark, and don't look like the smaller akoya oyster shells typically used in these wish pearl or pick-a-pearl operations.
You can learn more about Tahitian pearls here:
https://www.pearl-guide.com/forum/content.php?42-Tahitian-Pearls
"Natural" is a term that is sometimes used incorrectly to mean "real" or "not fake". But when referring to pearls, "natural" actually means "wild"-- that is, a pearl that occurs in a wild oyster or mollusk, completely apart from any culturing process.
The market for pearls is now a
cultured pearl market. There are very few natural pearls found any more.
The free pearl coupon is a clever marketing ploy; who wouldn't like a free pearl? But they are not about to give away valuable cultured Tahitian pearls. They make their profit by setting an inexpensive pearl in precious metal jewelry.
I'm sorry you got taken.
If you and she like the settings you bought, you could always have them re-set with real cultured Tahitian pearls. Of course that involves more expenditure but in the end at least you will actually have some very nice jewelry.