As per Timatic, the database used by airlines:
Passport exemptions: Passengers with an official photo ID proving nationality
issued to British nationals for travel between Ireland
(Rep.) and Great Britain and Northern Ireland only.
Ironically, no British photo ID other than the passport proves nationality.
In practice, however, virtually all airlines, including FlyBe, accept UK driving licences, as stated on FlyBe's website.
At Irish immigration, simply present your boarding pass and driving licence, and you should have no problems.
British citizens of obviously non-British origin (such as those with another country of birth as stated on the UK driving licence) should present a UK birth certificate in addition to their UK driving licence, as these documents combined prove identity and nationality.