This is indeed beyond the scope of actual knowledge for humans. I will speak some nonetheless.
Firstly, to suggest that Jesus had two natures on the Earth is essentially a superficial analysis. Humanity was created in the image of God. Jesus was the image of God. He was, by nature, the prototype human, before the incarnation. The Church is the female version of Jesus, so in a sense, one might say Jesus and the Church are of the same "species" and Jesus just became one of that species, which He already was.
"Species" is typically used to refer to created animals, so it sounds weird, but when you consider species' have a female and a male and that Christ and the Church are in that mold, you can see what I mean.
Edit: Perhaps to restate: Do we have to accept that God - in his omnipotence, must be able to create the logically impossible, in order to believe in something like the trinity?
What we consider logical is based upon that which God has created, and I would argue, what He has instituted. Him recreating things is perfectly within His authority, but I think that with Him there are promises He has made and He has otherwise made His intentions known and unchanging. His intentions unchanging isn't the same as His creation unchanging, but God has established the Earth forever (Psalm 104:5).
I don't think the Trinity is outside of our current logic. I could hypothesize plenty, but without me doing so, you can even see the idea of unity in the biblical worldview. Marriage is the idea of two people that are so unified that they are considered one. Additionally, literally, a human's DNA is an exact replica of the DNA of their parents.
Imagine if you were Jesus and you only had only a Father. He gave you His DNA, per se. You are an exact replica of Him now (see Hebrews 1). The question of whether Jesus and His Father are 1 person or 2 people is now a genuine one. Sonship is about accurate representation of parent(s).
I'm not suggesting that the Father is finite the way Jesus is. I'm suggesting that Jesus exactly reveals in a finite way, the infinite nature of the Father and the Holy Spirit. Are they three or are they one. Well it's kinda both. I'd recommend asking God when Jesus comes back.