My husband and I are based at a BoE. Our desks are across from each other, and we keep some of our less-used supplies there. We have 14 schools between us (11 elementary, 3 junior high). We each go to a different elementary school every day, save during the small blocks of junior high school time. The rules are that if a school is within 2 miles or so of our apartment, we go straight there in the morning because we can just walk or bike there. If the school is further than that, we go to the BoE first and then one of our co-workers drives us there and back. If we're scheduled until 3:00/3:30 at a close school (depending on which school it is), we can go straight home afterwards; otherwise we go to the BoE directly after school. We used to have an "office day" once a week every other month, but that was cut, so we've started keeping all of our materials at home. We also bought a printer and a laminator, which means we can do most of our lesson prep at home, too. Between all of this, we spend an average of 3-5 hours at the office a week during the school year. We do need to go to the office every day when we don't have classes (including summer, winter, and spring vacations), which provides lots of time to make materials and revise curricula.
Unfortunately, since we're the only two ALTs at the BoE, I can't give much advice on the omiyage situation. =/ I wouldn't worry too much about overloading them, though; BoEs are used to it. After all, in April, when the school year begins, every single new principal comes into the BoE with armloads of gifts. Within a few days the place looks like a sweet and senbei shop. And if nothing else, people will appreciate the thought.