Washington
Wow thats a loaded question if I ever saw one. Predicting the weather above treeline is really difficult at best. I believe though, that the route from Adams to Washington certainally stands to be harder then from Madison to Adams due to the terrain and exposure which I belive is more significant.
You must also take into account that if your camping at Grey knob that you need a fairly signifigent weather window to safely make the return trip and if your overnight gear is stashed at the camp, you leave little option then to return. Although carrieing all your overnight gear would make for a slower time it would give you more options for descending or bailing out if you had too. Personally, I wouldnt use grey knob as a base for Washington, but thats just me, not trying to ruin your itiniary.
I think its a good base for Adams, Jefferson and Madison but IMO Washington is to far for a prudent round trip in the winter, unless you get stellar weather all day, which is what it would proboly take you safely. Now if you carry all your gear and descend say Lions head, I think its a very doable trip as you have your equipment with you and that trip is very doable, sorry I cant remeber if you meant to return to Grey knob but thats mu humble opinion anyway.
P.S. Assessing the conditions at Edmonds col might help on the way to Washingtons summitt, but things can get ugly up there quick and you couldnt imo count on that weather to hold for the return trip if thats what you intended. You could expect anything from high winds, extreme wind chills, low visabiltiy, blowing snow, or very good conditions you must be prepared for all of the above. Be safe and remember turning back while hard, sometimes takes more guts then continueing on.