An AC is a heat pump too.
I just looked it up, and I can buy a heat pump for 200-400 euros (depending on desired output), installation is ~400 euros. Why are you paying 20-30x for something identical? This sounds like a price difference created by government behavior, like with solar panels and related hardware which seem to be significantly overpriced in north america.
It's a price difference created by market segmentation of heat pumps as a luxury product in the US, and the relative lack of qualified installers due to our under-investment in education in the trades.