I definitely think this is a job for photo editing software after, rather than being able to get it perfect while you take it. I have the same problem with teal (and a little with purple, but I find teal much worse). I try to get it as close as I can on the camera with the white balance, and then mess around in software to tweak things further.
Whatever you use to edit your photos, look around for some colour settings (colour balance and hue are my most used) and have a little play to see how they change things. Often with these frustrating colours everything else will look right, and it’s just one particular colour that looks off, so you’ll want the settings that separate colours out more so you don’t make everything else look wrong while you fix the green. Remember it’s normally better to change things a little at a time rather than do big jumps, or the colours can go wildly different. Sometimes one setting will get you a little closer, and you need another one to get it just right.
You definitely need to lighten up the green a little, because they don’t change much when they’re too dark. Sometimes I also find fixing the colour gives a tinge to the background (which is when it’s easier to have a pure white background), but obviously with your white centre that’s not easy, so I just had to lighten it a little so the greyish white didn’t look too greenish white.
Although it’s hard to tell if it’s right without being able to see the actual item, trying to guess what green it might be because even searching “emerald green” brings up lots of different greens