The version in Reynardson's Sussex Songs (tune and one verse) isn't the same as the Sussex Archaeological Journal one. It also quotes F E Sawyer's text from Journal of the British Archaeological Association 42, 1886, 324 (much the same as Rock's in SAJ) with his brief account of the custom, which I don't think is available online. I can do you a copy of that if it's of any interest.

The scan of notation at Google Books is quite hopeless. I'm afraid they tend to go for quantity rather than quality. I may be able to decipher it, though.

There's a further account (also Sussex), taken from G J Monson-Fitzjohn, Drinking Vessels of Bygone Days [1927], at

http://www.nicks.com.au/index.aspx?link_id=76.673

It refers to an unidentified piece in Notes and Queries, but doesn't add anything much to what we already know. Beware of the annoying pop-up window.