Wow, BBs will go through snowboarding goggles like bullets through butter. They are super-thin, brittle, and not designed to stop anything but fluffy snow. You are underestimating the power behind a BB. A BB can shoot straight through lab goggles.
Don't wear snowboarding goggles. Don't wear anything you haven't tested by shooting point-blank first.
When you are wearing your balaclava, you will also most likely fog through your goggles to the point where you can't see unless you have a pair with good ventilation or even a fan solution.
When looking for an eye solution, look for an ANSI z87.1
rating. This is the minimum rating you should accept.
Even with that rating, do a test. There are fake products out there.