Do post offices renew birth certificates?

No, they don't. And I don't think they ever will because this will require sensitive data to be made available to them and different sort of checks to be carried out to verify the identity of the individual. As it stands, even though they accept payment of utility bills like water, if there're any arrears on the bill which has been cleared before a new bill has been issued, they are unable to confirm the payment and ask you to go directly to a CWA office. Therefore they have basic or very limited information available to them.

So you'll definitely need to go to a Civil State Office.

civil status office,
actually at pamplemousses both civil status and post office found in the same building