Honestly, bliblical angels are already described like lovecraftian monstrosities beyond time and space. If you just want to ape the aesthetics of cosmic horror, lots of tentacles and wings and eyes and a morality that the human brain simple cannot comprehend will get you like 99% of the way there. But that's not actually cosmic horror, just wearing the skin of it.
"Now all my tales are based on the fundamental premise that common human laws and interests and emotions have no validity or significance in the vast cosmos-at-large." — H. P. Lovecraft
Cosmic Horror is based around the insignificance of man, that there is no grand plan or design, and the universe is vast cold and uncaring if not outright hostile to our existence (not because we are special, but rather because we aren't). It's the opposite of the divine inspiration. Paul on the road to Damascus glimpsed the full width and breadth of the universe and that God had a plan for each and every person and was uplifted by that realisation. The Cosmic Horror protagonist glimpses the full width and breadth of the universe and is broken by their insignificance in it and hopelessness before it.
The horror isn't in the monster, it's in what it represents: The mere existence of a cosmic horror monster is irrefutable proof of mankind's insignificance and the futility of existence. It's hard to achieve this with a traditional demon, as demons and angels presuppose mankind having a significant role. That our lives, our 'souls', have some meaning to them. I think the only way to achieve true cosmic horror with such a concept is to find a way to flip that on it's head, find a way to make what the demon represents terrifying in a way that renders all of human achievement meaningless. For example:
There was a God. There was a plan. The plan failed, and God left the universe to wither and decay. And He took the afterlife with him. There is no heaven left to ascend to. All souls go to hell or just disappear. The angels who remained became demons. In a world without the light of God to give purpose, they began torturing and tempting and corrupting humanity. But not out of malice or to test us, but in desperation. Perhaps, just perhaps, if they torture us enough, if they damn enough souls, if they cause enough pain, then maybe God still cares enough to return. Maybe they can bring Him back. Maybe he cares enough to stop them. They've been doing this for thousands of years. It hasn't worked yet.
It still puts humans on a pedestal, but the pedestal is cracked and broken and there's that sense it's doomed to crumble to dust. And I think you need that sense of hopelessness to be true to the concept of Cosmic Horror. In this example the disappearance of God is the cosmic horror aspect, and the demons are just the proof of it.