What's actually happening
Your heat pump water heater isn't malfunctioning. It's doing its one job: pulling heat out of the surrounding air and moving it into the water tank.
The byproduct is cold air. The unit blows out exhaust that's often in the 40s or low 50s. In a warm summer basement that feels like free air conditioning. In January it feels like someone left a window open.
An unfinished basement in winter might already sit at 55 to 62 degrees. Add a water heater pulling thousands of BTU per hour out of that air, and you can watch the temperature drop 5 to 8 degrees during a heating cycle.
So the question isn't 'how do I fix the cooling.' The cooling is permanent to how the machine works. The question is where that cold air goes and how hard the heat pump has to run.
Why yours is overcooling
Nearly every case comes down to one of three things: the mode it's set to, the size of the room, and whether the exhaust is ducted.
Most units ship in their most efficient mode, sometimes called eco or heat-pump-only. In that mode the compressor does all the work and never lets the electric elements help. Maximum efficiency, maximum room cooling.
Room size is the next culprit. These units want roughly 700 to 1,000 cubic feet of air to breathe. Drop one in a tight utility closet and it recirculates its own cold exhaust until the space feels like a meat locker.
Then there's ducting. Without a duct kit, all that cold air just falls to the floor and stays there. A duct sends it somewhere useful, or at least somewhere else.
Two smaller factors: a high temperature setpoint makes the unit run longer, and an oversized tank means more capacity than you need pulling heat out of the room every cycle.