Let us first begin with the first example.
- Nothing but champagne, now /_\ I am the boss.
In this elision of " that", there are at least three rules at work if an explanation is sought.
1) There's the use of discourse particle.(now/then/oh)
2) 'That' induced noun clause is used as an object.
3) The sentence is too much opinionated especially from
the use of personal pronoun(s)
Any of these rules is reason enough to get rid of "that"— a subordinating conjunction but not a relative pronoun. If you like to view this as an example of elision, better call it " gapping" in the absence of any better name.
_. _. _
- There was a farmer/_\ had a dog, and Bingo was his
name-o.
Such statement may be common and unique to a particular dialect but has no relevance to English as a universal language— more so when we omit a relative pronoun at the subject position. It has no relationship with our first example.