For instance, should I say: 1) "The book count is overwhelming" or 2) "(The?) Books count is overwhelming"? I am specifically interested in the version without "of" (as in brackets in the question).
In your sentences, "count" is the subject and "book" is an adjective. The verb and the definite article need to agree with "count" (i.e. without the adjective, you would say "The count is overwhelming," not "Count is overwhelming.").
There's no hard rule about whether a noun used as an adjective should be plural, but in most cases, the singular is used. "Book count" sounds much more natural than "books count" (LD).
The book count is overwhelming.