The simplest answer, as you already indicate, is the higher partial pressure of oxygen in the gas, leading to a higher 'solubility' at the interface of gas/liquid. The 'driving force' or deficit increases. But how do you intent to increase the inlet pressure? Generally this is done by increasing the top pressure, what increases pressure all over the height.

Secondary effects may be the initial bubble size and superficial gas velocity, but that depends on what you do: initial bubble size depends on gas velocity through the holes of your sparger/nozzle and also superficial gas velocity is gas volume flow dependent. So if you take care to keep this constant, there should be no serious change. Average bubble size depends heavenly on initial bubble size with low liquid height, say below 2 m. When higher, the effect becomes small enough to neglect. But a large increase in pressure (bars) does have (some) effect, but I assume that is not the case.

I think Rob already well explained the effect. One thing I'd like to point out is about the pressure difference between the inlet and the overall pressure in the tank. I think it's the pressure difference between the two that will affect the bubble size etc. And in most cases, to increase DO, the overall pressure is increased as well as the inlet pressure, so their effects on bubble size may cancel to some extent.

and here is some additional reading if you want:

https://www.eppendorf.com/product-media/doc/en/864241/Fermentors-Bioreactors_Protocol_034_BioFlo-320_Dissolved-Oxygen-Control-PID-Tuning-Cell-Culture-Applications.pdf

https://plygenind.com/how-to-control-dissolved-oxygen-do-in-bioreactor/