Well, no, although I think that it's unfair to say that BA's lowest fares are cheaper than bmi. A few samples of lowest fares on respective routes:

(fare / tfc / total)

LHR-EDI BD £4 £46.30 £50.30
LHR-EDI BA £22 £44.10 £66.10

LHR-DUB BD £10 £29.25 £39.25
LGW-DUB BA £32 £42.70 £74.70

The tiniest of BD's new tiny fares are really pretty cheap. The trouble is, 'tiny', as we've discovered, runs the gambit from £40 to £150.