King George V – Harry’s great great grandfather – limited titles within the royal family in 1917.
This means Harry and Meghan‘s first born, as a great-grandchild of the sovereign, is too far down the line of succession to be an HRH.
George V declared that: “the grandchildren of the sons of any such Sovereign in the direct male line (save only the eldest living son of the eldest son of the Prince of Wales) shall have and enjoy in all occasions the style and title enjoyed by the children of Dukes of these Our Realms.”
The eldest son and heir apparent of a duke can use one of his father’s lesser grade peerage titles by courtesy.
So a first son of Harry’s would become Earl of Dumbarton – one of the subsidiary titles Harry received from the Queen on the morning of his wedding.