Many people win four letter drs and that was a bunch of solicitors trying it on, three letters must be even more complicated, the only way I can see them having rights in a three letter is if they used to own it and displayed it on all their letterheads as such. Depends how many you have against you already and how much you stand to lose, if anything, roll with it, solicitors by their very nature are supposed to be obnoxious and irritating, that's their job.
Ignore them or reply with "it's a three letter, if you have rights then so does everyone else which pretty much means, it's generic and you have no rights, have fun" Ignore them, don't reveal your hand in any correspondence as you're just doing their job for them.
You'll find most big companies have solicitors inhouse and it costs them nothing to tell them to start writing letters as they're on a full salary and contracted anyway so them sending letters from a solicitor is no big deal, just means the director or advertising bod can give it to that dept. to deal with, in many cases I should think it's accounts you'll be hearing from next offering you money.
It costs them nothing to use an inhouse solicitor, an american company used one in London to write to me over here, I still got them to agree to pay money but I don't think they ever sent it so I sold it to someone else.