I don't care how the rest of the world spells or pronounces Mackie/MacKay/McKee/etc. My intention here is to document the works of Harry Lauder, and if the earliest published works of Harry Lauder used the spelling Mackie, that's how I'm going to spell it. Now let's examine the evidence.

Here's the British Library catalog data for the sheet music of Mr John Mackie.

In case that link goes dead, I have copied their catalog entry:

System number – 4469684
Cataloguing level – Minimal record
Author - personal – Lauder, Harry, Sir, 1870-1950.
Title – Mr. John Mackie. [Song.] Written, composed ... by H. Lauder.
Publisher/year – London : Francis, Day & Hunter, 1903.
Physical descr. – fol.
General note – With a separate voice part.
Holdings (All) – Details
Shelfmark – H.1309.b.(25.)

The sheet music is also held by University of Oxford and Syracuse University, which also use the spelling Mr. John Mackie.

Then there's the University of California, Santa Barbara, which shows that the 2 wax cylinders published (by Edison) in 1911 and 1927 used the spelling Mr. John Mackie, but the 78-rpm disks published in 1903 (Gramophone and Zonophone), 1909 (Victor), 1919 (His Master's Voice), and 1920 (Victor) used the spelling Mr. John MacKay (or Mackay). Go figure.

By the way, Lauder also sometimes uses the word Scotch where people today would insist on Scottish or Scots. Again, my intention is not to conform to current accepted usage, but merely to document Lauder's actual usage.