Folksingers who have sung this all have used 'Van Dieman's'; writers of the time also spelled it that way.
There is only confusion to be had if the spelling is corrected.

See this book (Google):
Henry Walker Parker, 1834, Van Dieman's Land: Its rise, progress and present state with advice to emigrants, London, J. Cross, and Simpkin and Marshall; Oliver and Boyd, Edinburgh; and Currie and Co, Dublin. 244 pages plus appendix.

also:
"Van Dieman's Land," in Historical Narratives of Early Canada.
http://www.uppercanadahistory.ca/tt/tt11.html