Peggy has long said [have that] meaning either [(you) have that] or [(I
want to) have that] as the pragmatic context makes sufficiently clear.
In a typical scenario today, Peggy was unnecessarily specific in her
utterance, thus. Peggy frequently plunks some object (a book or toy) in
one’s lap, says “have that” and indicates her desire to lap-sit. Today
she placed a doll in my lap and said /***/. When I asked “Who have the
doll?” she responded [get up], and coming around my knee, made
clear it was she who should “get up.”

Relevance: Peggy here strung together two utterances which we would
recognize as ‘phrases.’ [have doll] was unnecessarily specific. I
interpret its use as a sure sign that the utterance “have that” has
become a two-element phrase with one variable. Contrast “get up” with
the contrary “get down” (Peggy interprets both adequately) which may
be more easily conceived as two related idioms with a common
utterance core (/***/), whose commonality may be more accidental
than meaningful (as perceived by the child).