OP: Why does the Holy Spirit not know who Jesus is?
I believe this question is flawed from the start because it assumes as a premise that the Holy Spirit does not know Jesus, and then seeks an explanation for that premise. I claim that the premise is wrong to begin with, in light of what 1 Corinthians 2:10-13 (ESV) says:
10 these things God has revealed to us through the Spirit. For the Spirit searches everything, even the depths of God. 11 For who knows a person's thoughts except the spirit of that person, which is in him? So also no one comprehends the thoughts of God except the Spirit of God. 12 Now we have received not the spirit of the world, but the Spirit who is from God, that we might understand the things freely given us by God. 13 And we impart this in words not taught by human wisdom but taught by the Spirit, interpreting spiritual truths to those who are spiritual.
Basically, the Spirit of God knows everything that God knows. Therefore, if God knows who Jesus is, it logically follows that the Spirit of God also knows who Jesus is. Also notice that the Spirit of God has cognitive capabilities, because it comprehends the thoughts of God (it understands things), which presumably includes things that are way beyond the comprehension of any human or even angelic mind. In other words, you could argue that the Holy Spirit has an omniscient mind. And also, that it is alive (this is a reasonable idea to accept considering that all instances of spirits in the Bible, either good or evil, are living conscious beings, and also considering that spirits are the 'breath of life', i.e. spirits are what make stuff alive).
In short, you could argue that the Spirit of God (1) has an omniscient mind and (2) is alive. Those two attributes alone sound very much like attributes of a person. So, though not a definite proof, at least they can be viewed as compelling reasons to justify a belief in the personhood of the Holy Spirit.
OP: Has Jesus forgotten the co-equal, eternal God, the Holy Spirit here?
Not necessarily, I think there is still room to think that Jesus is implicitly including the Holy Spirit in the conversation. First of all, as we just saw, the Spirit of God knows everything that God knows, so restating this fact every single time would be unnecessary or sound redundant, and maybe Jesus didn't deem it necessary here. Secondly, the Holy Spirit was indeed part of the scene. If you look at the verse right before your quote, you will see that the Holy Spirit is the one who inspired Jesus to say what he said:
Luke 10:21-22 (ESV):
21 In that same hour he rejoiced in the Holy Spirit and said, “I thank you, Father, Lord of heaven and earth, that you have hidden these things from the wise and understanding and revealed them to little children; yes, Father, for such was your gracious will. 22 All things have been handed over to me by my Father, and no one knows who the Son is except the Father, or who the Father is except the Son and anyone to whom the Son chooses to reveal him.”
Moreover, in Luke 4:16-21 (ESV) Jesus explicitly acknowledges the anointing of the Holy Spirit on his life and messianic ministry, so he was definitely aware, and claiming otherwise (that Jesus somehow "forgot" about the Holy Spirit) would not make much sense:
16 And he came to Nazareth, where he had been brought up. And as was
his custom, he went to the synagogue on the Sabbath day, and he stood
up to read. 17 And the scroll of the prophet Isaiah was given to him.
He unrolled the scroll and found the place where it was written,
18 “The Spirit of the Lord is upon me,
because he has anointed me
to proclaim good news to the poor.
He has sent me to proclaim liberty to the captives
and recovering of sight to the blind,
to set at liberty those who are oppressed,
19 to proclaim the year of the Lord's favor.”
20 And he rolled up the scroll and gave it back to the attendant and
sat down. And the eyes of all in the synagogue were fixed on him. 21
And he began to say to them, “Today this Scripture has been fulfilled
in your hearing.”
OP: Or, is this a confirmation that there is no 'person' called Holy Spirit?
I already offered a couple of reasons why you might think the Holy Spirit has personhood: (1) it (very very likely) has a(n omniscient) mind and (2) it (very very likely) is alive, two attributes that, to the best of my knowledge, only apply simultaneously to persons. You can find other reasons people have compiled to believe in the personhood of the Holy Spirit here: What is the biblical basis for the personhood of the Holy Spirit?. Again, these are not definite proofs, but you will hardly ever see a formal, logical, mathematical proof for anything in Scriptural matters. It's unfortunate, but that's the reality in any discipline that lacks a standard, logical, mathematical formalism.
OP: The disciples also suggest this latter option is the case, the so-titled 'Holy Spirit' is another term for the Spirit of the Father. (Luke 12:12, Matthew 10:20)
Yes, I agree. In fact, both the Son and the Father send the Holy Spirit (e.g. see Who sends the Holy Spirit? The Son or the Father?). However, I wouldn't be so quick to claim that the Holy Spirit and the Father are exactly the same. To be honest, things get philosophically complicated when we try to offer a formal ontological description of the nature of the Father, the Son and the Holy Spirit, and I won't do that here. Yet, I still think there are compelling reasons to think that the Holy Spirit and the Father are in some respect distinct from each other (they are distinct entities, beings, persons, whatever you wanna call it). You can find examples of verses pointing to this conclusion in the following question: How do Biblical Unitarians understand verses that appear to identify God (the Father) and the Holy Spirit as different beings/entities?.
In sum, I would say there are compelling reasons to believe that:
- the Holy Spirit is omniscient (and therefore knows who Jesus is).
- the Holy Spirit has attributes of a person.
- the Holy Spirit is not the same as the Father nor the Son (they are different beings, entities, or whatever word or philosophical term you may prefer to use).