I would follow the GPS directions and time it to be on the Beltway between the hours of 10AM and 2PM. We lived in the area for many years and found that we could make it South during this time period without a lot of delays unless there was an accident.
Any other way such as Frederick, MD would be a long way out your way. Just take your time and stay in your lane. People drive the beltway all the time and don't have major problems.
You could take the 301 bridge and catch I-64 West and connect with I-81 South that way. Longer, but less traffic. 301 bridge is narrow and a little spooky to drive over.
In summation, there is no good way to catch I-81 from the Bay Bridge area. Good luck.
