How to apply Henry's law down the sea?
What does it even mean to calculate or measure solubility of something down the sea?

Why, there is a precise and well-defined meaning, and it is hinged on equilibrium between the gas and the solution, which supposedly should be established at the said conditions. Our imaginary gas bubble, like everything else, must withstand the immense pressure of all those layers of water up above.

So no, the partial pressure is not just the atmospheric one.

Deep abyss of the seas is undersaturated in all gases, including oxygen, and that by a wide margin.