Hi Mrsspen,
I'm not a military person, but I've been Googling your enquiry, and offer the following:
Wikipedia here:
http://en.wikipedia.org/wiki/Order_of_battle_for_the_Battle_of_the_Sommeprovides information which I may be misinterpreting, so judge for yourself.
It appears from this that 1 Battalion Rifle Brigade were a regular army unit, i.e. not one of Kitchener's callup.
They formed part of the fourth division of the army which fought at the battles of Albert, 1-13 Jul 1916, and at the Battle of Le Transloy, 1-18 Oct 1918, which appeared to involve capturing Eaucourt L'Abbaye and le Sars, and attacking the Butte de Warlencourt. According to CWGC William died on the 18th, sadly the final day of the battle.
Entering le sars into google maps and zooming a little shows these three places to be all within a mile of each other. Must be worth a trip one day. I know a few people who have made this kind of pilgrimage, and they all report it as profoundly moving.
I see others have been busy while I was looking this up! You have a concern about the unit being active. Wikipedia shows the fourth division in battle, and not in reserve. No doubt a War Diary could confirm this.
Good luck with your research.