A prospective customer is trying to contact one of the team members through a non-company e-mail to verify that we are the "real thing", what do you think about this situation?

If it’s not very difficult for the CEO to respond from the academic email address, it could be a low-lift way of verifying the affiliation. Just have that person on the team bring the CEO back onto the email thread / conversation saying that the CEO is the point person for business development at the company. It would be odd for them to dictate this process. Just assert that the CEO is the team lead on this internally.

You’ll want to make clear who is the account manager and all requests should go through he/she.

Having an advisory board on your website also helps so potential clients can reach them as references.