I posted this earlier. It might be more pertinent here. Tomlin and similar were very common Christian names in previous centuries in Britain. Proof, simple...just try looking up the surname 'Tomlinson' i.e., Tomlin's son' in a telephone directory, so any use of the name in a song shouldn't be linked to another use of it without other corroboratory evidence.

The 17th century usages probably refer back to stories about a mischievous sprite with that name, which may or may not have had influence on the ballad and the humorous/racist song.

The 1627 usage yes very likely influenced the 18th century writer of the long ballad. (IMO).