The 3 replies so far are all ignoring a key line in the OP's text, "I tried using a different faucet, which doesn't seem to do the snow globe effect." If it were the water, it should be common to all taps. To my thinking, the only difference between the tap the OP is using that is providing the precipitate and the other taps is the hose connecting his sprayer to his kitchen fixture. Perhaps the texture of the inside of the hose provides a much greater surface area that facilitates crystallization of the hard water compounds in his water and now some of that crud is being mechanically washed off the inside surface of the hose into the water stream.
CY: The hose is only a couple years old -- as I wrote. The soft copper in the floor is gosh knows how old.
The OP should add a few tablespoons of household white vinegar to a pop bottle with the precipitate. If it is a carbonate salt, the increased acidity from the vinegar should dissolve at least some of the precipitate.
CY: Good test for carbonates.
However, part of the OP's text is puzzling. He says that the precipitate becomes evident when the water warms up. He should be observing the opposite. Calcium and magnesium salts are more soluble in warm water and less soluble in colder water. The precipitate should be most evident when the ice first melts, and less evident as the water warms up and some or all of the precipitate dissolves.
CY: The way the OP figures, the solids ppt out when frozen, and are visible when the frozen ppt thaws, and becomes clear.
This makes me wonder if perhaps the stuff is not a chemical precipitate, but rather microbial crud. Another test would be to let a different pop bottle with the precipitate (one that has not had vinegar added) sit for a few days at room temperature. If the precipitate substantially increases in quantity, I'd question whether or not he is looking at bacterial or mold colonies that are multiplying. If so, once again, the seed crud is probably living on the inside surface of the hose.
CY: Worth checking. I'll have the OP check that question, and let you know what he finds.
The solution to the problem might be to figure out if the crud is chemical or microbial. Then disconnect the hose and give it a good soaking and cleaning (or buy a replacement). If the crud is chemical, use undiluted vinegar. If it is microbial, use some clorox, 1/2 cup to a gallon of water.
CY: I've been wondering if I ought rig some way to pump some clorox or vinegar or both into the cold water line. Not sure how I'd do that, but something will come to mind. I mean, uh, if the OP should do that.