Allan, as you say, the Sate's name changed from The United Kingdom of Great Britain by ADDING Ireland to it. There were three kingdoms in 1707, but only two united, the Scottish and the English making the United Kingdom of Great Britain.
The crowns weren't united in 1603 as is commonly stated. All three kingdoms were independent of each other but shared a common king until 1707 when only Scotland and England became politically united, Ireland still being outside this political union as a separate Kingdom with a separate parliament.
If a new United KIngdom is created to go by the name of Great Britain, then surely one can logically say that this new State is a United Kingdom of Great Britian? (pre the joining of Ireland).