Skipness - yes bmir were a full subsidiary of bmi until sold not so long ago.
Rebranding is expensive - but the bmi name as a whole is tarnished - it's viewed as a defunct airline bought by BA. If bmir want to save the cost of rebranding that might be very sensible. If there is good reason to rebrand, then it should be to a new identity that can be shaped by current bmir management, and not try to clothe themselves in the mantle of the past.