Always at night or go around. I like rt 301n/s to avoid DC. Where are you coming from and going to? Other wise drive at night. Post your starting location some forum members might have an alternate route.
Most people don't drive through DC, unless they want to see the sites in DC (memorials, museums, Capital, White House, etc.). If you're just passing through with no intent to go into the city, you'll likely be taking the Washington Beltway (I495) to I95 in VA. You can be sure that this route will be a nightmare during rush hour on M-F. On weekends, Sundays are lighter than Saturdays, but both days are typically pretty reasonable. I'd keep an eye out for accidents and construction prior to hitting these roads and consider alternate routes, if possible.
Taking route 301 is about 50 miles longer if you're coming south on I95... yes, it's much more tame, but on the weekend, I'd take the shorter route.
You are kind of stuck. Last time I went that way I did take the Francis Scott Key Bridge (toll) in Baltimore instead of the tunnels, much less traffic. I would also take the Belt way 495 S over the Woodrow Wilson bridge vice going through DC. There was a lot of construction south of the bridge to 95 South. Good luck.
This takes you around the top of Baltimore which is good because it is shorter and avoids the bridge tolls and the tunnels which you're prohibited from entering with LPG tanks. If you're doing this on a Saturday, I wouldn't worry about it at all... I'd just check the traffic prior to hitting these areas and make changes accordingly.