I just wish there were some record of his enlistment!The National Park Service database lists him as
Fish, WilliamSide:Union Unit Name:17 New York Infantry Regiment:17 State:New York Function:Infantry Company:H Rankrivate Description:Buried in National Cemetery Capture Date: Capture Site: Alternate Name: Remarks:H 10 NY ; H 17 NY INFANTRY ; NOT FOUND IN 10 NY INFANTRY  OR 17 NY INFANTRY 
Since there is some confusion, apparently, about what he regiment was (different sources say different things), I'm inclined to think that this is the brother of Lester. The records are only as good as the handwriting of the person writing them and the interpretive abilities of the person reading them.
I'll dig a bit and get back to you, but an April death is consistent with the 270 grave number.