I would stay away from the idea of saying that Jesus is somehow two persons no matter how one defines person. Such usage of the word ‘person’ I believe will only lead to confusion, errors concerning the person of Jesus and who Jesus is and possible outright heresy. I say this even though I think you believe as the Church teaches that Jesus is solely one divine person and your more or less broad interpretation of the word ‘person.’ Concerning Jesus and the truth that Jesus is one divine person in two natures, our theology and usage of language needs to be as precise as possible. The eternal Son of God assumed the nature of man but not the person of man or a human person. Simply put, the Word became flesh and dwelt among us; God became flesh, God took on the form of human nature in the likeness of a man.
Concerning whether Jesus has two consciousnesses and being that the divine person of the Eternal Word subsists in two natures since the incarnation, i.e., divine and human, I think it could probably be said that Jesus has a divine ‘consciousness’ and a human ‘consciousness’ according to each nature Jesus possesses. However, the bearer or subject of these two natures from which results as it were the ‘two consciousnesses’ is none other than one divine person, the eternal Son of God. The human nature of Jesus from which results the human ‘consciousness’ as it were of Jesus is the human nature and human consciousness of none other than the eternal person of the Word. The individual or person experiencing the human consciousness in Christ is none other than the Eternal Word. In fact, whatever can be said about the human nature of Christ belongs to the eternal divine person, the Son of God. So in the Litany of the Sacred Heart of Jesus, we say ‘heart of Jesus, of infinite majesty, have mercy on us.’ This means that the physical heart in the body of Jesus is the heart of God because the Eternal Word assumed a human nature in his person. The humanity of Jesus belongs to the divine person of Jesus who is the eternal Son of God. At the same time, we understand that the physical heart of Jesus is predicated of the human nature of Christ which he assumed in time. I know it can get tricky of what can be properly said of Jesus since he is one divine person in two complete and unconfused natures. But the complete human nature that the Eternal Son of God assumed does not mean that He assumed an individual or person of human nature otherwise we would have the heresy of two persons in Christ. The person in Christ as man is the second person of the Trinity.