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Like an engineer told me recently, the image of AC is biased in most of us towards the electricity-sucking devices of the previous century. Contemporary AC tech is to 1980s-90s AC what LED lightbulbs are to incandescent ones.
Moreover, the very same units are often also super efficient heating devices that embarrass their gas/oil burning counterparts.
Assuming the manufacturer has bothered to make the tiny modification necessary to turn AC into a heat pump and then not charge you 50% markup for the privilege.
Heat pumps are still stupid expensive.
https://youtu.be/_hAuKtoRxJI?si=3LO40Ih8XGKmbWmC
Tldr, gouging exists, but people regularly also get systems that are way bigger than they need, which drives costs up by many multiples.
It's an even bigger problem in countries that don't traditionally have air conditioning systems because basically no one knows how to size these things.
But even disregarding that, a similarly sized AC, (similar to whatever heat pump you are quoted for, which as you say will probably be too big) will still be more expensive than an AC on its own even though it's basically the same thing.