There would need to fundamental changes in legislation to allow that to happen - and really? Given all the outcry about the privatisation of public services, you think the public would be happy with a fat-cat led Commercial Company managing state land?
Sorry, just seeing this now NN. But my understanding is that this already happens. The crown estates are managed by an independent company. The profits go to the treasury, who pay the sovereign grant from those profits, but keep the majority for state spending.
If we dismantled the monarchy, the only difference would be that the Sovereign grant doesn't get paid.