DisplayCal will host a webpage that you can connect your phone to and test the display on any device with a probe.
I don’t know if the iPhone will accept a ICC correction lut. But you can at least adjust the factory settings in it to get the least bad results.
If there is a similar photo with the eyes closed, it certainly is possible to cut the the eyes out and paste them over in Photoshop.
https://youtu.be/ZYPhKxisatA?si=iIlWSsXR5czFfxes
If they offered “deep photoshoping” as a feature of your package, I would hope this is within their skill set and time budget.
As the eyes are the focus of the photo, it needs to be done well or it will ruin the photo.