I should really do some research before I sound off!

First, the Scottish Travellers from who the 'Tam Lin' versions were collected were Betsy Johnston and Willie Whyte. The texts are very fragmentary but I think that I can hear Bert's tune in both versions. You can hear these tantalising fragments on the CD 'The Muckle Sangs' (Greentrax Records, CDTRAX 9005, 1992). The frustrating thing about this CD is that the notes by Hamish Henderson talk in very general terms about Scottish balladry but tell us nothing specific about these two remarkable recordings. In addition, neither singer names the male protagonist of their respective fragments (so we get no clue about the validity of the 'Tamberlaine' form of the name). One thing that does strike me, though, is that these recordings are from ORAL tradition. I have heard it said that 'Tam Lin' is only known from print - these two recordings suggest that this view is mistaken.

Second, there is a recording of Bert singing his version available on CD. The recording in question is 'Classic A.L.Lloyd' (Fellside FECD98, 1994). This is a live recording made at a Runcorn folk club in 1972. Bert can be heard introducing the ballad as "Tam Lin or Young Tambling". The protagonist's name doesn't occur until some verses into the ballad when Bert sings 'Tambling' ("She hadn't pulled a herb, a herb, a herb but barely one, when up then spoke the Young Tambling says, Margaret leave it alone..."). In the next verse I'm sure that I can hear another syllable ("Come tell me now, Young Tam-BER-lin(g?) she said, if a mortal man you be ...").
In my memory of the occasion at which I first heard him sing the ballad, in Peterborough, 4 or 5 years earlier than the Runcorn performance, I'm convinced that he used the 3 syllable form (and the last syllable was closer to 'lane'/'laine' than 'lin'/'ling'.

So, I think that the simple explanation is that Bert often used a 3-syllable version of the name because it scanned better when used in the Traveller derived tune; the similarity to the name of a medieval Tartar conqueror is just an unfortunate coincidence.

Finally (and I've gone on far too long already!), Child's G version uses a 3-syllable name, ie. 'Tam-a-line' and his H and I versions use the forms 'Tam Lane' and 'Tamlane' respectively.