Could BMed be booked via the BA website and did BMed use 'Birdseed' callsigns per chance?
From the time of the BA franchise being signed all bookings were done through BA. The callsign stayed as "BeeMed" as the aircraft remained on the AOC of the owner, British Mediterranean Airways. Once they were bought out by BMI the AOC was no longer needed and the aircraft moved across to BMI and used "Midland" as the callsign.