Fort Greene Coffee house Recommendation
Not technically in Fort Greene, (but near BAM,) is the flying saucer cafe on Atlantic between Nevins & 3rd. Call ahead though... I heard they were changing owners so they may be temporarily closed. Haven't been in a long time.
Otherwise, you're going to have to walk up DeKalb, and none of the coffee houses in Fort Greene proper are either low key or comfortable.
There are a few on Dekalb. Bitter Sweet has really great coffee, but it's fairly small. Right next door is Catherine's which is your best bet. Great coffee and food. It has a really comfortable atmosphere and you can hang out there as long as you want. The owner is really nice. Both are on Dekalb between Cumberland and Carlton. Only a short walk to BAM.