There are two parts to this question that need to be addressed in order to answer you:
Is it the case that “Avalokiteshvara was a cosmic being representing the compassion of all Buddhas” or was he a bodhisattva/mahasattva who was present when the Buddha taught?”
And, does “Avalokiteshvara speak through the power of the Buddha or is just that Avalokiteshvara's compassion is "part" of the Buddha since he is indeed fully enlightened?”
In the first matter, clearly, in many Mahayana sutras Avalokitasvara is present as the Buddha taught. But was he a ‘cosmic being’ that presumably would not be present in that way? Although he has and continues to be depicted that way, there is some evidence that this is an inappropriate characterization that leads those that hear him being described as the “lord of the world” to also believe that he would not be present in the way that you and I might have been present if we were alive when the Buddha taught. According to Alexander Studholme in his book, “The Origins of Om Manipadme Hum: A Study of the Karandavyuha Sutra:”
"Avalokitasvara eventually became Avalokiteśvara, it seems likely, due
once again to the force of another folk interpretation of the name,
here based upon the identification of the bodhisattva as a lokeśvara.
This is a generic term meaning, literally, “lord” (-īśvara) “of the
world” (loka-), applied to a wide range of supernormal beings in
Indian religious thought. Bearing in mind, then, that very few people
would ever have seen the name of the bodhisattva in written form, it
is quite understandable that the pronunciation of the -asvara ending
should have slipped to the homophonic -eśvara, thereby producing a
name—Avalokiteśvara—that would actually have seemed a more appropriate
title for a being understood to be a great lokeśvara."
There is no need to confuse the bodhisattva/mahasattva Avalokitasvara as a “lord of the world.” And if that is let go of, then we can see this bodhisattva/mahasattva for what he was: a realized being and the perfect embodiment of “Great Compassion” (Mahākarunā), which is the activity or function of Emptiness.
In his description of his meditative practice leading to his enlightenment, he says:
First, because I did not listen to sounds and instead contemplated the
listener within, I can now hear the cries of suffering beings
throughout the ten directions, and I can bring about their liberation.
This power that he accomplished by discarding sound at this point in his practice in order to contemplate the meditator, which is to say, meditating upon the nature of hearing—given the presence of uncaused, unborn, and unending inner spontaneous sound—Avalokitasvara disengaged himself from both sense organs and their perceptions, enabling the direct realization of this all-embracing (Mahākaruṇā) Buddha naturing of all living beings and actual things. This is the nature of ‘mind’, which is the necessary direct insight for further progress along the path.
But the most important point of his statement is this: using the practice that is described by Avalokitasvara very quickly results in what I describe as an alchemical change in the practitioner, which is normally only accomplished much, much later along the path to enlightenment. Here at this very early stage in his practice, Avalokitasvara attests to being able to hear the cries of suffering beings and respond to them so that they can be liberated from their suffering. This is the result of having used inner spontaneous sounds, which are nothing other than the intrinsic Emptiness. They are not a manifested phenomenon of Emptiness, but rather the resonances or reverberations of the activity of Emptiness manifesting (dharmata) the world and all beings and things.
You see, by initially meditating on these sounds and using them to break free of sense perceptions and senses in order to have a direct experience of this intrinsic naturing, your hearing faculty is very quickly liberated from, and no longer constrained to, sensed sound. Hearing the cries of suffering beings is not the same as our mundane hearing sounds of crying. I cannot stress this point enough, but only those that have accomplished this hearing will truly understand this.
Although this might seem to be off the point, I bring his description and my commentary on it in order to point out the answer to your question: “does Avalokiteshvara speak through the power of the Buddha or is just that Avalokiteshvara's compassion is ‘part’ of the Buddha since he is indeed fully enlightened?”
Avalokitasvara does speak through the power of the Buddha, but only in the sense that the Buddha is the expression of Emptiness here in this world, as is Avalokitasvara. The result of enlightenment is the removal of our ignorance that blocks the true way the world ‘works’ from our view and understanding. Once that ignorance is let go, like removing blinders from one’s eyes, we realize directly that all beings, all teachings, all compassion, is the direct functioning of Emptiness.