There is no data suggesting that the Titanic (known for its first class service) did not issue formal hats to it’s live lobsters. It is however known that the Titanic did not source it’s lobster from Mexico (the bandito style hats seen in the gif) and probably had a mix of European and American lobster aboard.

As it was a first class service it can be expected that the lobsters were given top hats to along with their tails.