This is more of theoretical question, not a practical, but it might be useful for some (we already have a system). I live around Auckland, NZ where heat pumps are very popular, typically ran for about 4-5 months a year. They are useful for about 2-4 weeks a year as air conditioner as well, but not absolutely essential.
Most popular system here is split system with some offering fireplace style indoor unit - which makes total sense as heat travels up so your split unit takes way longer to heat up the room.
To me tho sounds it might be simpler to just install window units mounted level to the floor. They are typically much cheaper and don't require extra piping for refrigerant.
BTU's aside, am I missing something why this wouldn't or couldn't be done?