1. A system that also provides cooling isn’t allowed under the heat pump government grant.
2. You might actually need planning permission for this
3. Especially in a humid environment like the uk, you can end up with condensation and damp issues, which you really don’t want in an underfloor slab install
If you also want cooling, better to go an air-air based system (aka air conditioning)
The amount of cooling this way is limited. Though the nice thing is that it tends to keep the house cool. There is no cold airflow.
With respect to government grants. Sometimes you can buy units that can only heat to get the grant. But a simple change enables cooling after the unit is installed.
Might provide some benefits on really hot days, but it’s still uncomfortably warm for working in.
There is no reason to set all units to the same value in the entire country.
In general, if you want to save money, you should set your aircon to the highest temperature you can tolerate. We typically operate at 26-28C at home here in Singapore, and our dew point is typically between 24-26C.
The modest reduction in air temperature from 30-33C outside feels amplified, because the aircon also reduces humidity. We also rely heavily on ceiling fans.
That said, I'm guessing 22 (or 18 or whatever) is the lowest temperature you can have on the pipes; that wouldn't translate to 22 in the house.
If their working fluid is water, you are right. But most aircons work like fridges, and their pipes don't transport 'cold' by literally having a cold fluid, but by having an ambient temperature fluid that they turn into a gas inside your interior aircon unit (and then later turn back from a gas into a liquid on the outside unit).
The actual pipes are always at pretty much ambient temperatures.
You only end up with condensation issues, if you cool below the dew point.
Here in very humid Singapore our dew point is typically at about 24-26C. If you set your aircon to no less than 26C here, you never have any problems with condensation.
(Any old weather reporting website or app will tell the dew point. Apparently right now as I am writing this comments, it's 12C in London.)