The unit is the easy part. The emitters decide, and the answer is different for each of your two floors.
Radiators cannot usefully cool. A radiator moves heat mostly by convection, driven by warm air rising off it. Run cold water through it and there is no convection to speak of — cold air sinks and pools at the floor, the radiator does very little, and you get a cold patch instead of a cool room. The output when cooling is a small fraction of its heating output. So upstairs, where you actually have the problem, radiators are not the route.
Underfloor can cool, within limits, and the limit is condensation. A cooled floor works: it is a huge surface area and it takes heat out by radiation and a little convection. But the output is modest — a floor can only be so cold before it is unpleasant to stand on, and that caps how much heat you can remove. It handles taking the edge off a warm day. It does not handle a heatwave.
Now the condensation, which is what your installer was gesturing at and is the thing to take seriously. Air holds a certain amount of moisture, and if any surface falls below the dew point of the room air, water condenses on it. Cool a floor below the dew point and you get a wet floor — slippery, and over time a genuine problem for the floor covering and anything under it.
This is why proper underfloor cooling is not just "run the pump backwards":
- You need humidity measurement in the rooms, not just temperature, and a controller that calculates the dew point and limits the flow temperature to stay safely above it.
- The safe minimum flow temperature therefore moves with the weather. On a humid day it is high and you get very little cooling — which is, annoyingly, exactly when you want it most.
- Ventilation matters more than you expect. Open windows on a humid day and you are importing moisture and pushing the dew point up.
Also check whether it is even enabled. Plenty of reversible units are commissioned with cooling disabled in the installer settings, because enabling it without dew point control is a callback waiting to happen. Sometimes it needs a hardware option, a valve or a sensor. Ask specifically whether the unit supports it, whether it is enabled, and what dew point protection is in place.
For the upstairs problem specifically, the realistic options are fan coil units — which blow air over a chilled coil, handle condensate through a drain, and actually dehumidify — or accepting that a separate air conditioning unit does this job far better. Fan coils can run off the same heat pump, which is the elegant answer if you are prepared to fit them.
What I would not do is run cold water through existing radiators and hope. It does not work, and any condensation happens on pipework in places you cannot see.