Seems like acting on privileged imformation for personal gain. I think that would be a very risky idea. OTOH advocating for the same role from within the comp

Seems like acting on privileged imformation for personal gain. I think that would be a very risky idea. OTOH advocating for the same role from within the company is probably a way to repair the project without pre-burning your bridges. You could still leverage those connections to switch companies (basically, show the client thar you want to formally represent them).