The basis of this is the upapurana named Brihaddharma, which allowed Brahmins of Bengal to eat fish. The english translation of the book is here.
Please also see the following from the book, History of Bengali people (page 364) by Ray.
Another of Bengal's scholars, did the same; quoting two verses from
the Vishnu Purana; he tried to show that, except for a few holy days,
on no day was the consumption of fish or meat an abomination.
According to the Brhaddharma Purana, rohita, saphan, (pufi or
saphari), saphara and other white, scaly fish were edible by Brahmans.
Having given a list of animal and vegetable oils and fats Jimutavihana
mentioned the oil of hilsa fish and described its various uses. It
would seem that in ancient times as much as today hilsa fish was one
of the favourite foods of the Bengalis and that its oil was used in a
number of ways. Not all fish, however, was edible by Brahmanas; they
were forbidden to eat any fish that lived in mud-holes, whose mouth
and head resembled a snake's (such as the bana fish), or which had no
scales. There were also prohibitions against the consumption of rotten
and dried fish, although the writer of the Tikasarvasva, Sarvinanda,
said that the people of Bengal were very fond of eating dried fish.
Such meat as snail, crab, rooster, breast of stork, duck, water-fowl,
camel, cow and swine were completely forbidden, at least in the
society regulated by Brahmanical law.