If our universe is so complex that it couldn't have happened without a designer, how is it possible that God does not have a designer?
Something has to be eternal and causeless, otherwise you end up with an infinite regress, which is a severe logical problem. It seems you yourself recognize this. So, either God (an eternal causeless mind) or the universe (an intricate and complex array of matter and energy) is eternal. We can go to science to see which is more likely.
If God is special in that he's eternal and causeless, why is it not that our universe may be eternal and causeless?
Because the universe (matter, energy, space,
and time) had an absolute beginning about 13.7 billion years ago in the big bang. This is the overwhelming consensus of modern cosmology. Given that science is the epistemological foundation of naturalism, to reject this consensus is a violation of your own epistemology. So, the universe is not eternal.
On top of all this, the universe is intricately finely tuned to allow the evolution of intelligent life. If any of the fundamental physical constants of the cosmos were off by even a tiny fraction of a percentile, life and even stars could not exist. These physical constants are not physically necessary as a result of physical laws, they just happen to be programmed into the universe at the moment of the big bang. This is indicative of purposeful design.
The question then becomes this; which is more likely to be the causeless cause responsible for this finely tuned universe, an eternal mind (God) or an infinite set of unfathomably complex universes with random designs somehow interacting with each other and existing for no apparent reason, by chance producing a universe like ours in an infinite cosmic dice game? (multiverse hypothesis)
Following Okhams Razor, God is the simplest explanation to explain the observable fact that we inhabit a finite and finely tuned universe
If God is beyond human comprehension, why is it not just that our universe is beyond human comprehension?
I think it is in many ways, but not entirely. Actually, the very fact that we can learn real truths about this universe is itself indicative of God. Under a naturalistic frame-work, where both ourselves and the universe exist without reason or purpose, it seems incredibly unlikely that a species like us would evolve and actually be able to learn about the laws of the cosmos at all. Yet we do really know things about the universe, because we use this knowledge for practical applications that work.
Further, where does our sense of natural beauty come from? Why is nature beautiful, in a naturalistic frame work? I can see no evolutionary benefit to perception of beauty in nature.
However, if God does exist and wanted to reveal some things to us about himself by means of his created order, then it makes sense that he'd equip us with the ability to learn real truths about the universe and to perceive it as beautiful.
"For since the creation of the world His invisible attributes, His eternal power and divine nature, have been clearly seen, being understood through what has been made, so that they are without excuse."
- Romans 1:20