Sorry BD = bmi. I should have made that clearer.

No way to access lounges in arriving terminal before transferring to departing terminal, with the exception of the designated arrival lounge (which you dont qualify for access anyway) and this would require clearing immigration and a landside terminal transfer.

Fast track is mostly for immigration rather than flight connection centre. However there is a lane for BA & QF premium pax at flight connection centre security check. You wont qualify for that either, but sometimes the line bouncer will direct some pax to use it (if no BA/QF pax and the other line is enormous).