Standalone usage of "tid" from tidbits
Solution 1:
The OED has four entries for tid, two as a noun, one as a verb and one as an adjective. None has the meaning ‘whole part of something’. The only use associated with tid-bit is obsolete. Even then it seems to be little more than a shortened form, as this definition by Johnson suggests:
Tid, adj. (tydder, Saxon), tender; soft; nice‥Titbi′t (properly tidbit; tid, tender, and bit), nice bit; nice food.
So, no, I think the writer was just playing with the word. Not that there’s anything wrong with that.