They wish to recreate the Troubles, have the army rescale its presence, and have everyone act as if it's the 1980s over again. By repeating their name unquestioningly, the BBC is abetting their aim. Today's Wall St Journal refers to a "IRA renegades". The Sun refers to "a cowardly 'Real IRA' gang". Best of all, The Guardian denies them any formal name and introduces them only as "dissident republican paramilitaries".
No one would seriously argue that the BBC's decision is political - it is just the unintended consequence of a house style enforced unthinkingly by a sprawling bureaucracy. Companies spotted this flaw in the journalistic system a while back, and ordered newspapers to play around with the capitalisation (such as "easyJet"). They complied. Then groups like Invesco demanded that their whole name be put in capital letters - not because it was an acronym, they because they wanted it to stand out on the page.
Terrorists are just as capable of using spin. And the BBC should know better than to fall for it.