Take the bus to Disney's Hollywood Studios. From there, you'll be able to get a boat to the BC, or, if the boats aren't running yet in the morning, a bus. I think the first boat leaves DHS shortly after 8.
You should allow at least 1 hour to get to your destination; so if you want to be at the restaurant at 8:15 for your 8:30 reservation, be at the bus stop at AoA by 7:15. When you board the bus, tell the driver that you are going to the BC for breakfast.
It's likely you'll arrive at the resort in much less than an hour, so be prepared to spend some time walking around, or have some activity to keep the kids (if any) busy.
And yes, Epcot's International Gateway opens about the same time as the main gate. Once you get in, some parts of World Showcase may be roped off, but you'll be able to get to Future World no problem.