The sentence "Of milk I have but none" is grammatically correct and means you have no milk. But is being used in its sense of merely or only. One could of course say "Of milk I have but little" and the sentence remains correct when little is replaced by none.
Edit: My answer above addresses the grammatical correctness of "but none" as used in the question's example. My answer does not suggest that one should actually so use it or that it has been so used. However, here's a slightly-awry example of such use:
What the two doctors describe as necessary for the citizenry of a republic, our countrymen have but none ... If we now insist on republicanism, no matter whether we can establish it or not, are we going to follow the steps of France, ...
Note, the vast majority of google book links for have but none are irrelevant, being artifacts from comma-punctuated phrasing as in "More intelligent readers you may have, but none more grateful than ...".
The phrase merely none, on the other hand, has numerous
google book links where merely none is used precisely for the purpose of meaning none at all. Many of the links are repeated references to identical passages in various printings, and many are punctuation artifacts, so only a few dozen of the several hundred links are relevant, but in those few dozen cases, merely none means none at all.