Great job cleaning the machine.
Yes, it is possible to clean around the bobbin area. Go to my Youtube channel (see below - sorry, since the changes on this board you can't just click on it for the link) and you will find full instructions on how to clean a machine. I have done two videos, each 13 minutes long. Part 1 has had far more views than part 2 which makes me wonder why people don't watch part 2 - don't they want to see the end result?
If you unscrew the screw by the little ejector tab at the front of the bobbin case you can actually get the bobbin holder out and clean it thoroughly. If you find it awkward getting the parts out and back in, unscrew the needle plate to give easier access.
Let me know how you get on!