Do you also agree that that activity would still be 'organised' if you remove one (however significant) element, whatever actual form it takes?

Would such a green-socialist system be based on a price-profit mechanism in which the factors of production were organized (e.g. by worker-owned corporations, cooperatives, entrepreneurs, state-owned businesses, etc.) to obtain a surplus (or profit) from society in return for the goods produced by this organization?

In short, no.

First, the 'green' part of green-socialist means that it's not society alone that is considered. It's now widely recognised that for any economic activity, you could consider externalities. But one could go further and specifically consider external depreciation, that is loss of value: from the lowering of real estate prices for homes near a new highway through air pollution to the loss of uniqueness (as a worth on its own) in the form of species going extinct. Now,

1. quantifying these externalities would be very difficult and contentious, and may require new technologies (like electronic money was) to function;

2. but what really matters is not the actual value but that these have to be considered by default;

3. when you take externalities into account, it is very likely that overall surplus is impossible [if it can be reduced to the 2nd Law of Thermodynamics...] and we end up discovering that we are managing a decline.

Now, the question could be re-phrased by talking about "surplus" as what remains after fixed wages, even if that too belongs to workers. However, why exclude the possibility of a company/enterprise doing without wages? So, we could consider things in terms of one workers' share on the income. Which the worker would of course wish to increase -- but if it is a bit more in one year and a bit less in another, is that a catastrophe?

But, the concept of growth gets muddied further if we also consider that workers could collectively decide to re-invest more of the income minus expenses, or that workers could collectively reject schemes that would increase one (remaining) workers' share by reducing the workforce and increasing productivity. So from price-profit mechanism, we are back to a price system and just "wish for a good life" as the basis of economic activity.

_{*Lunatic*, n.
One whose delusions are out of fashion.}