What your state allows
Updated
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What the federal rule covers
Milk and milk products in final package form intended for direct human consumption, delivered into interstate commerce or sold. Pasteurized, or made from pasteurized ingredients. That is a clear line and it is drawn at the state border.
Inside a state, it varies enormously
Some states permit retail sale of raw milk, some permit it only at the farm, some permit herd-share arrangements, and some prohibit sale for human consumption entirely. There is no pattern to infer and no safe assumption to carry across a border.
Dairy licensing is usually its own permission
Separate from a general food premises license, often with its own inspection schedule and its own equipment requirements. Ask about it as a dairy question rather than as a food business question, because the two processes frequently sit in different parts of the same department.
Cottage food almost never covers this
A product that requires refrigeration for safety is outside most cottage food lists by definition, and a dairy product is the clearest case of one. Treat that as the starting assumption and confirm it rather than hoping.
Record who told you what
Name, date, what you described and what they said. Guidance gets restated and staff change, and a specific conversation you can point at is a different position from a recollection.
Plant-based may sit somewhere else entirely
An oat or nut beverage is not a milk product, so it may fall under general food rules rather than dairy licensing even when made in the same room. That can cut either way, and it is worth asking as its own question rather than assuming it follows the dairy answer.