You might want to point out to them the case of one Martin O'Neil then.
I have this arguement with my suposedly non-exsistant northern Irish future sister-in-law. Martin O'Neil is a former northern Irish defender, who played for his country for many years, and with a name like O'Neil, there is little of English to him.
BUT, he also has an MBE (Member of the British Empire), which in my opinion, must suggest he has taken on board and embrassed himself as "British". My brother's fiancee completly disagree's with this and insists he is indeed Irish and NOT British.
It is a point of view of course, I actually had a storming row in a New York hostel of all places with 2 rather burley Irish gentlemen (who strangly were big fans of cricket) about irish independence, suggesting Britian was hit "in the crotch" when it was down by the irish was non-too-popular.
Of course, there are MANY, MANY who would suggest they were NOT british, certainly those who switched their lights on to guide the luftwaffe to English towns would agree with you. But again, a lot of them probably DIDN'T think of themselves as British, but they fought for the British empire, their country was occupied by them, and in general, when your country is seen (at that time anyway) as prt of Britain, and you fight FOR Britain, surely there is no more example you can have of them being British?.
Irish people today will always vhermently disagree when you suggest they are British, because now they are not, but once they were, just like once the Scotti tribe in Scotland was Irish, but call a Scot Irish and see what you get.