I agree that the name was being bandied about after 14 April 1912.
We have the poster of unknown date and origin, showing a Cunard ship.
There's a silly tale in The New York Times about Gigantic, which will be 1,000 feet long and sport a cricket pitch and other unlikely amenities.
There are figures in Lloyd's Weekly News and also in The Deathless Story of the Titanic. Gigantic is listed as building. She's to be 1,000 feet long and 112 feet wide. She's to be 50,000 GRT, which is obviously stupid. Such a ship would have to be something at least 80,000 GRT.
Anybody who knew anything about H & W would have known that 1,000 foot ships could not be built in its facilities. Another sign of fantasy!
I see no real reason why the sequence of Greek names should have been followed. There was a big gap between the orders for the ships. The first two were ordered on 31 July 1908. The order for Britannic followed on 20 June 1911.
The first intelligent and accurate reference to the third ship that I can find dates from 11 December 1911. She's described as being much like the earlier two, but a little wider. No name is given.
The earliest reference to Britannic seems to be on 31 May 1912 in The New York Times. This may have given the impression of a hasty change of name following the sinking.
I have an important document on this but I was allowed to use it in my e-book under strict conditions. It's good evidence that the name was never going to be Gigantic.