Firstly welcome to the BA board on FlyerTalk. I agree with you that your options are limited from EMA and BHX. I think your idea of driving to MAN and doing a day return trip to LHR is a good option. Search Google Flights for options. A quick search reveals that you can do a day trip on Saturday 18 May on BA for £94 for the flight. You only need the cheapest TP earning fare in each direction and could use Avios to reduce the cost even further.
The other option is to do a weekend trip somewhere with the family.