There are a few things that we should look at here: Genre, Book, Psalm, Sentence Features, and the context of all God's story. I think you've gotten most of that, but I'll try laying it out. For this discussion, I won't reference the NT usage of the text to avoid any sort of circular reasoning.
It is poetry - as you've tagged. In poetry, we do not expect the same sort of literalness as in other genres. We can expect figurative language and exaggerations in poetry. In Song of Songs we read in chapter 6, verse four that his beloved is like an army with banners. In the verse before, we learn that the two beloveds belong to one another - this wouldn't be true by the literal legal mores of the time - that characters are saying how it feels. It is poetry and that's allowed in poetry, and today is sort of the point of poetry.
The Book is the Psalms - a book of psalms, that is a certain sort of poetry. It has been called "Jesus' prayerbook". We can be confident in our identifying Psalm 22 as poetry and thus causing us to expect figurative language. But what is the book about? Bible Project accurately describes it as a book about lament and praise, but one that never forgets God's faithfulness.
The book of Psalms is supposed to be an encouragement to hope and faithfulness is response to God's faithfulness. For there to be literal forsakenness would conflict with the point of the book.
This is Psalm 22. How often is figurative language used here? And what is the point of this Psalm? In verse six, we read "But I am a worm and not a man, scorned by everyone, despised by the people.", scorned by everyone is figurative language since most of the world did not know the Psalmist existed, and "But I am a worm and not a man," is certainly figurative language. So we can say with confidence, that in this psalm the psalmist uses figurative language to portray the misery.
(Mike Winger actually has an interesting discussion of this worm.)
The first two thirds of the Psalm is filled with imagery asthe Psalmist talks about his sufferings. But then starting with verse 22 the figurative language stops, and the praises of God begins. This looks like a deliberate technique from the author to go from figurative sufferings - including that attention-grabbing first line - to vivid and literal descriptions of the glorious future. And that contrast is the point of the Psalm.
The word forsake is a word often used with God in the Old Testament, but almost always (there is one exception) about how God does not forsake his people (with is more than can be said for his people with God). So not only is the sentiment jarring, but the precise word jars too, however the feeling does not jar.
The exception is Deuteronomy 31:17, but that has a clear condition of forsakenness - to go after other Gods. This is something that the Psalmist gives no hint of doing.
The psalm is also written in the first person. What this sentence is doing is grabbing our attention, and connecting to us on the subjective experience level. However, by using feelings that we know cannot be so it leads us away from those feelings and to the wonders of God. Which is the point of the Psalm.
God's Big Story
So, we know that as a Psalm that it can be figurative language. And we know that Psalm 22 does use figurative language for sufferings in contrast to the literal glory. But are we certain that this first verse is not a literal starter that pre-introduces the animalistic imagery later? Well other thoughts in this first verse like "so far from my cries of anguish?" seem to dispute God's attributes, such as his omnipresence.
Also, these are questions. He does not make figurative blasphemous statements. He is asking the questions that people ask, and he is answering them with God's faithfulness.
And that is the point of the verse, the psalm, the book and the Bible. He is forsaken figuratively for "For the LORD your God is a merciful God; he will not abandon or destroy you or forget the covenant with your ancestors, which he confirmed to them by oath.".