Do It Yourself

This will work, in theory, and if you're willing to use a lot of water. It's probably a bad idea.

Heating one kilogram of water by one degree Celsius without phase transitions (freezing/melting, evaporating/condensing) takes 1 kilocalorie of energy. That's roughly 4 kilojoules aka kilowattseconds, or 0.0012 kWh.

Thus, to get 1.2 kW of cooling, which is about half of what those tiny portable air conditioners promise, at a 10 degree temperature difference, you'd need 100 liters of water per hour. If water costs $0.40 per 100 liters, and electricity cost $0.40 per kWh, an air conditioner (using about 0.4 kW of electricity to pump 1.2 kW of heat) will be a lot cheaper, and that's ignoring the power you might need to run the pumps and fan on your solution (all of which you get back as heat!)

Unless the water in the loop is below the dew point, you also won't get any dehumidification. This is actually more important than cooling, and a big reason why air conditioned rooms feel so much better (sitting in the shade in 40° C dry weather would be unpleasant but fine, at 100% humidity, it would be reliably fatal regardless of fitness).

If you're building new, look into:

• proper insulation
• insulation and windows that optimize for the right thing for your climate (in countries like Germany, I suspect windows are optimized to let as much heat in and as little out as possible, which saves heating costs in winter and turns apartments into hellholes in summer)
• passive cooling paint and panels - I don't know if they're commercially available and in a practically usable state yet.
• solar to power the AC
• swamp coolers aka evaporative cooling (the split kind that evaporates water outside). Downside is they use water (which actually is lost - evaporated), so if you're in a drought prone area where water is restricted or expensive they might not be the best choice. Also, it has to be actually dry (low humidity) when it's hot. Get actual, local climate data, not gut feeling. Check if there are commonly used commercial solutions, possibly combined with actual A/C (very common for industrial scale setup, not sure if common for home setups).
• regular air conditioning. I'm assuming you're trying to build a house to live in, not an art. Economies of scale mean that going with suboptimal but standard solutions almost always beats custom hacks. If you have the same brand of AC as everyone around you, the repairman will know how to repair it, will have spares, will know how to design it so it is sufficient for your house, etc. - if you build something yourself, you will be the only one who can maintain it.
• ceiling panels - these cool the room by running cold water (generated using normal A/C heat pumps) through pipes/panels under the ceiling. The upside is that they also remove radiant heat, the room feels about two degrees colder than it is thanks to this (look up "wet bulb globe temperature" for a rabbit hole). The downside is that they can't dehumidify and actually stop working in high humidity when you'd need them the most: if you run water colder than the dew point through them, it'd condense and start dripping all over your stuff, so it shuts down or limits how cold the water can be (and thus how much it cools). Consider them as an addition only if they're common and installers are familiar with them.

In the end, you're building a new building, so you now have a chance to do everything right using modern but already proven technology. I wouldn't DIY anything critical and hard to change like this. Remember, you're trying to find the best (likely: cheapest in the long term while meeting your reliability requirements) solution that will solve your problem. There's a very high chance that's simply "add more A/C and solar according to what's locally available". And that's fine. There's nothing bad about that.

I wouldn't, for example, try to build with different materials than locally common, even if those were "better" by some metric. That often doesn't give you a better house, that gives you a unique house, and unique can be a nightmare.