KLM has 3 main waves of European flights (2 morning, 1 evening), as well as a more spread out fourth wave in the afternoon.
The most popular wave for UK/Ireland flights is the third wave, with 17 out of 18 airports (including BHD) operating during this period. The first wave includes 14 airports, and the second wave includes 15 airports.
The early morning wave of KLM European flights depart the UK/Ireland between 05:55-06:30, and arrive in Amsterdam between 08:10-09:00. A 06:30 departure from BHD would arrive in AMS at 09:05. However, the final European arrivals in this wave are at 09:15, meaning a 06:30 departure from BHD would be feasible.
If however it is impossible for KLM to operate out of BHD during the early morning wave, it is completely feasible to add a second daily flight during the second morning wave, with departures between 09:00-10:00, and arrivals in AMS between 11:00-12:00. Several UK/Ireland airports have their first KLM departure during the second wave.
Operating during the second wave would allow for almost the same number of connections to European, Asian and North American destinations as if BHD had a first wave departure.