Captain Bligh was certainly Bligh of The Bounty, showing great courage when set adrift in an open boat by the mutineers, though at that time he was only a Commanding Lieutenant.
But there are other reasons why Mrs Fitzhardinge might have been impressed by him. Bligh was later Captain of HMS Providence on Cook's last and fatal voyage to the Pacific. And he was Captain Bligh when he sailed in HMS Porpoise to New South Wales in 1805 as the new Governor of New South Wales with orders to put down the officers' rum trade.