Looks like you just need an additional permit (same immihelp source):
Permit Granting Authorities:
MHA & FRROS
All Indian Missions abroad
Chief Immigration Officer, Chennai
Immigration Officer, Port Blair
If arriving by air, permission to stay up to 30 days is issued on arrival at the airport in Port Blair.
If arriving by sea, permit is ...
According to the website of the Andaman and Nicobar government, you will be able to receive the Restricted Area Permit on arrival at Port Blair Airport. The website also mentions that you'll be able to apply for the permit at all Indian missions abroad.
While I would assume that you will be able to board your flight to Port Blair from Kolkata without any ...