My question is about that term "the Police". Would these have been ordinary members of the battalion but with special duties? In other words, is it reasonable to deduce that he was one of the men killed by "an unlucky shell"?
Yes I think that is who is being referred to. Any Military Police (later Royal Military Police) would have been stationed further to the rear, often manning checkpoints at road junctions. It is unlikely anyone in the battalion would know their whereabouts or if they became casualties.
On the other hand the Regimental Police would have been located relatively close to the front line. Their duties included checking the identities and authorisation of anyone entering or leaving the battalion rear area and being initially responsible for holding and searching any enemy prisoners of war brought in by members of the battalion. Members of the Regimental Police tended to be more mature and trustworthy soldiers. They were ofter given the local rank of lance corporal in order to have authority over other members of the battalion. They didn't have any 'special' powers, for instance of arrest, which the Military Police did. Members of the Regimental Police would wear a black brassard with the letters RP in red on it.
