I'd assume that British-Irish would be more likely to be used of someone born on, or settled in, the island of Great Britain, who is of Irish nationality or origin. Rather like London Irish.

For people in Northern Ireland "British Irish" it's not really a suitable term, since Ireland is a separate island. They may be citizens of the United Kingdom, but that is a "United Kingdom of Great Britain and Northern Ireland". The "Great" in Great Britain, it's just a historical way of distinguishing it from Brittany in French - "Grand Bretagne" as opposed to "Bretagne".