It has to do with the temperature of the light source. In your case the print that you are looking at acts differently under the different light sources. In the outside light the print doesn’t absorb all the green part of the light spectrum instead it reflects it. Which is why you are seeing the greenish tone.
What is your final mounting environment for the print? inside or outside? If it’s inside I would just print the job as it is, seeing that according to you you get a good result if viewed indoors. But if the final place is outside, check your profiles. The profile that you are using is incorrect for your final result, if you have the possibility change the settings. If you aren’t so hype with the ICC profiles I suggest that you tweak your curves just a tine bit, lowering the yellows and cyan. And then try your luck!
PS. I’m assuming that the original final is correct and doesn’t has green laid on top of the blacks.
Simon