Logan ticket counters do not call anyone to check on LTC's etc. Never had any issues with that happening, nor anyone ever asking if I'm licensed, but I do have a Mass LTC anyway, so wouldn't have any problems.
That said, I would advise flying into Manchester instead without a MA LTC, just to be on the safe side. Logan isn't like NYC or Newark and they don't seem to jam up travelers passing through, but YMMV. I'm unclear on RI handgun possession laws, so you'd have to research that.