You have lots of options, so it partly depends on your preferences. (I also have no idea which gates you can go in. My kids go to preschool across George Mason, and I know there’s a gate at G. Mason and 50, so that’s where I’m taking you).
My personal route would probably be this one. I haven’t ridden the new trail on the north/west side of 50 around Courthouse – it might be better to take the new trail on the south/east side and cross at Pershing. I just don’t like Pershing between Washington Blvd and Glebe (it’s lumpy and has annoying drivers, compared to the neighborhood route), but you might be fine on it. You can also shave a little distance by taking Oakland Street, which connects to the sidewalks around 50 and Glebe, then taking the “trail” along 50 between Glebe and FSI, but it’s just not as pleasant as going to G. Mason.
I’m sure there will be other opinions. As a general rule of thumb, pretty much any route you find between Rosslyn and FSI will be fine, with the possible exception of riding on Glebe for any amount of time.