Does the statement, "I do not want to rely upon Allah," uttered in a state of extreme anger, constitute apostasy that expels one from the faith and necessitates the pronouncement of the Two Testimonies?

This is a terrible statement, and there is a difference of opinion regarding whether it constitutes disbelief (kufr). The more authoritative view is that it does not, by itself, constitute disbelief, unless one intended to belittle the act of worship of reliance on God (tawakkul (definition)).

Anger does not lift responsibility or accountability, unless it overwhelms a person's mind to the point that they do not know what they are saying.

It is obligatory to repent and seek forgiveness for what was said. The basic principle is that a Muslim remains Muslim, and their Islam is not invalidated except by certainty of their falling into disbelief.