I have travelled this route with BA many times and not dream of bringing a passport with me. My UK driving licence showing a UK place of birth has never been a problem. I wouldn't bring a birth certificate, not has their been any need for a printed boarding pass. The mobile one is fine if asked for.

I do get the impression the border agents in Dublin do look a bit more suspiciously on a driving licence presumably as it is easier to forge, but it has never caused an issue. In terms of questioning I don't recall ever been asked any more than where I've just flown from. Perhaps in the response they are subtly making an assumption from the accent as to where you are from, but maybe that is overthinking it.

One thing to be aware of is that as you can't use the e-gates, you have to queue for a manned booth, so you need to add in extra time. I recall one occasion where it took me over an hour to get to the front of the queue, but that is not typical.

UK airports seem to treat the Common Travel Area more seriously than Dublin, so it is as easy as arriving on a domestic flight, albeit at Heathrow you land at the further away gates A22/A23 with the long corridor to avoid passport control.

In the unlikely event your flight from Heathrow is cancelled whilst you are airside and you cannot get another flight that day exiting Heathrow is slightly more painful. I've only experienced that once (when on a domestic flight) and I don't think the border agent was too happy that I didn't have a passport on me and only had a driving licence. After what seemed like a lot of questioning (including from what I recall some nonsensical ones like how long I'd been in the UK when my journey started and ended in the UK), and a lot of tapping on his keyboard he let me through to exit the airport.