My experience when I flew to DUB was that at the boarding gate at LHR no ID was checked (just like a domestic), on the return they were checking ID on boarding in DUB (everyone appeared to be using a passport).
Officially BA should accept any photo ID for this route. There's no legal need for a passport and passports are not checked on arrival into London.
I would use the driving licence. That said ideally you should get it in writing from BA that you don't need a passport for this route.
I can't find a link on ba.com specifically mentioning Dublin, but the route used to be operated by bmi prior to BA taking them over. You can still view the
bmi photo ID policy.
Acceptable forms of photographic ID
Travel solely between the Republic of Ireland and the United Kingdom.
Acceptable forms of photographicID include:
Valid passport
Valid photographic driver's licence
Valid photographic EU/EEA or Swiss national identity card
Valid armed forces identity card
Valid police warrant card/badge
Valid airport employees security identity pass
Valid business photographic ID (Dublin only)
Citizens of Ireland and the UK under the age of 16, do not need a photo ID, if travelling with their parent/guardian.
Citizens of countries other than Ireland and the UK must produce a valid passport and visa where applicable, or national identity card (EEA & Swiss Citizens), for travel between Ireland and the United Kingdom.
I doubt the policy has changed, but it's probably safest if you can get confirmed by BA directly.
Note: Aer Lingus seems to have a
very similar policy to what bmi did, so that to me makes it more likely BA have kept the bmi policy, but if you're worried you could always fly them instead if you don't get an official response from BA.