Two of my books were by MacLysaught (The Surnames of Ireland, p. 103 and p. 165; More Irish Families, p. 34 linking Biggar, Bickerstaff, MacGivern, and Montgomery in Co. Down.)
Other references linking the names:
The Scotch-Irish, Vol. II, p. 439 by Hanna, references another book "History of Lands and Their Owners in Galloway" by P.H. MacKerlie, and lists landowners in Galloway Scotland with Irish surnames MacGivern as well as Biggar.
The Book of Ulster Surnames, p. 195, by Robert Bell, links Montgomery and MacGivern families in Ulster. (not Biggar)
I'm sure you're right - MacLysaught must have used unattributed earlier sources.I first got interested in the Biggar connection when we were at a local highland games and I bought a Hall of Names genealogy for McGivern linking the name to Biggar. On the back is a long bibliography including early sources.I have unsuccessfully attempted to find some of the sources but appreciate any help you could give. If you are interested I could scan it in and attach it in an email so please send your email address.