West Britain might not have caught on, but West Brits abound.The term 'British Isles' is over 2000 years old, and pre-dates the existence of either Ireland or Britain as states by many centuries. Ptolemy's Geographia refers to the Bretanikai nēsoi,where 'nesoi' is Greek for 'Islands', and similar expressions - sometimes with an initial P- instead of B- - are found in earlier works.
The earliest known use of the expression 'Great Britain' to refer to the largest of the British Isles comes from the 12th century, where it was written in Latin as Britannia major, or in Norman French as grant Bretaigne. The term was indeed used to distinguish Great Britain from Brittany in France, rather than from Ireland. Britanny was referred to as Britannia minor, Bretagne menur or meindre Bretaigne in the same 12th-century sources. However, 'Great Britain' remained a fairly obscure term until 1604, when King James VI/I chose to describe himself as "King of Great Britain" rather than "King of England and Scotland".
Ireland has never been referred to as 'Little Britain'. For a time in the 18th century when it was fashionable to call Scotland 'North Britain', a few people jokingly called Ireland 'West Britain'; but that never caught on.