No, if it was a perfectly calibrated world it wouldn't be necessary at all, don't you see? The colours you edited on your calibrated screen would be the same colours they see on their calibrated screen. No grey card necessary.
No, I think you can achieve something nice. I'd make the background a radial gradient - pure black at the outside, and very very dark grey in the middle. So that the tonal shift is very subtle.
Both of them will work, you'll just need to spend some very patient time fixing the warm cast.
I'd say that we'll have to abandon the Screen method. The masking method it will have to be.
You really messed up your lighting, didn't you?
The practice doll photo you took the other day was really well lit. The actual baby photo has horrid mixed lighting.
Hmmm ... no, that's not right, is it?
We need to find a way to make it look like the ground drops slightly, around the body.
Also, what is with the weird straight vertical line on the left side of that closeup?