Yes you can keep the battery connected, just be very very gentle!!! Don't force anything. Maybe even wear latex/nitrile gloves so you don't get anything oily.
I think someone spilled coffee allllllllllll over my parent's interior. I even had to disassemble the iDrive controller and the traction control/driving mode buttons. The coffee would stick the traction control button down and then you wouldn't be able to select between driving modes. There was a tiny bit on the bottom left side by the gas gauge, so I had to wipe the inside of the lens and even on the cluster itself. Just be VERY gentle, even with a brand new quality microfiber I put very small scratches in it (nothing noticeable and it was still a net gain over the splash marks)