I believe a child needs "Home" no matter how international the lifestyle thrust upon them, otherwise they can become like a pendulum without a weight. The very clever Head of our school explains about his multi-lingual pupils that it is above all essential they master profoundly ONE language as it is via language that we think, reason, grasp concepts whether mathematical/science or otherwise, -- so one language in all its richness.
If I were you I'd seek the work link with Berlin via networks, conferences, shows, time there, co-projects, rather than move there. Keep Stockholm as Home.
As for the French, my view is if you do not speak with the child as maman in French from birth, and school them in the French system (Lycee Stockholm?) then they cannot truly hope to have all French career options open as an adult. That is not the same for other Euro countries and languages.
That said, in Asia for example it is quite normal for a child to be trilingual, via school, parents and tutoring. The difference is the parents are often trilingual in the same languages which is not your case.
On a final note, I know several international families who do the 3/4 languages intensively through Primary by whichever means (tutors, parents, nanny, grandparents to live in, school...) and then from 11 years old Secondary just switch fully to English as that it what they are rightly anticipating for the children's Uni and work beyond that.