Since the terms 'Briton', 'Britain', etc., seem to be exclusionary towards the Irish and other non-Britons, what would can we use instead?

As per usual, I have used the Internet, Oxford English Dictionary, etc., but, alas, it was all to no end, as the only thing about which I could read with respect to this issue was the disputes that have been held over this divisive issue, e.g. some people's claiming, typically Britons, that the Irish are X and not Y, etc.

The answer depends on what you're trying to denote. For example, I usually prefer to write about "citizens of the UK" even though UK law uses the term "British citizen." For something that excludes Ireland (and Northern Ireland), I might use "Great Britain," although I would be careful about the possibility that readers might understand that to refer to the entire UK.

To refer to Great Britain and Ireland collectively, I would probably say just that, or possibly Britain and Ireland. In some contexts, I might refer to the Common Travel Area, but that comprises additional territories (channel islands and Mann) that you may wish to exclude.

In short, the situation is complex, so there is no single term that works everywhere.