Use milk instead of water to prevent this. I have no idea why this works
Basic chemistry. (All baking is an exercise in applied chemistry.) Aqueous liquids (e.g., water) are mutually incompatible with lipids (fats) or hydrocarbons (e.g., mineral oils). In plain language, oil and water don't stay mixed without a lot of help. That's why lithographic printing (e.g., photo-offset lithography) works: printer's ink is a thick, oily substance, similar to artist's oil paint. The stone or plate is first treated so that water will fully wet it, and then an image is placed on it in wax, grease, varnish, or some other substance that repels water. On the press, water is applied to the plate, which saturates the non-image areas, so they repel the ink, then the ink is applied, sticking only to the dry image areas.
That is also why a simple oil-and-vinegar salad dressing starts to separate almost immediately. Mayonnaise, on the other hand, contains eggs, which act as an emulsifier
It's also why we use soap (another emulsifier) to wash with: soap is formed by boiling fat (e.g., beef tallow) with a strong base (e.g., sodium hydroxide), until they react to form a compound whose molecules have one end compatible with oils, and one end compatible with water.
Milk is an emulsion of butterfat and water, held together by another emulsifier, a sugar called lactose.
My hunch is that something in your frosting recipe is creating an unstable emulsion, and something in your food coloring is somehow prying the emulsion apart.
Personally, I've never had frosting break on me, and given that in various recipes, I've replaced significant amounts of milk with either strawberry jam or maple syrup, they had every right to do so. But I've always used 100% butter (specifically ordinary salted sweet butter) for the fat, I've always started by mixing the butter and sugar, before adding anything else, and I've always been fairly sparing with the food coloring.
And I don't think using 1% milk is part of the problem: with most of the butterfat missing, it should have excess lactose, and therefore more
emulsifying potency than whole milk.