No, it doesn’t originally come from a Bobbie Burns poem.
Nice investigation and appreciated by this Scot.
However, I don’t know if it was a typo or just a wrong assumption, but Robert Burns is rarely referred to as ’Bobby’. If you want to use the familiar version of his name, then use ‘Robbie’ or ‘Robby’ (I guess?)like most non-Scots English speakers.
However, in Scotland he is known to all as Rabbie Burns, the Scots variant of Robert as a nickname.