The LH and BD booking systems are based around the same (LH) system. The BD system has been a complete mess recently, because they have changed to a new "low cost" system of pricing flights as single flights.
The LH system in theory picks up any booking made in the Gallileo system (used by both BD and LH) that includes your M&M number. Sometimes this does happen, sometimes it does not.
The LH and BD reservations systems are not directly linked (your Infoflyway username and password will not let you access your profile on the BD website).
I would not trust what you see on the LH website - it is probably adding a lot of the details itself! I would be surprised if your tickets do not go to the address you specified to BD.
Finally, you do not need to reserve over the telephone to let them have your SEN number ahead of time. You can make the booking on the flybmi.com website (with the discount), and receive your PNR reference. Then call BD and ask them to add your SEN number - you get the discount, and they know about you in advance!