An in place water exchange should work fine if the fill water is colder than the pool water. The key is to keep the water still so it doesn't mix much while you are doing the exchange and of course the pump needs to be turned off. I would recommend this method over pumping out water and then refilling since it would be less likely for the pool to raise out of the ground should you get a heavy rain or have a high water table.

However, I would check with your local water district to see if there are times of the year where your fill water has lower calcium. In our area, surface water is used more during the winter than the summer so the fill water will have lower calcium in the winter than the summer. This means that if water is replaced during the winter, less will be needed.

Why does the fill water have to be colder?

Cold water is heavier, so it should stay on the bottom instead of mixing with the warmer (old) water (as far as I understand it).

Correct! The larger the water temperature difference, the higher the convection currents and the quicker the two bodies of water will mix. If the water temps are close to one another, it probably isn't such a big deal and you won't get much mixing. This time of year, tap water should be colder than the pool water but it depends on your location.

A submersible pump shouldn't require much explanation. Screw a garden hose to the outlet, drop it in the pool, put the hose where you want it and plug it in.

Easiest way to get a siphon going is to push the hose in slowly until it's all submerged and all the air has been pushed out. Then stick your thumb over the end to plug it and haul it out and get it below the water level as fast as possible. Then drag it where you want it to run out. As long as the hose outlet is lower than the water in the pool, it will flow. Don't let the pool end ever get unsubmerged or you lose the siphon. Your vacuum hose will flow better than a garden hose, but it's not likely as long.