As it is known that the 1st Iowa Cavalry (USA) victims were first relocated to Mexico and then Jefferson Barracks, it might be assumed that the requested men are also located there. The burial locations for the 1st Iowa Cav soldiers at Jefferson Barracks are identified, but I cannot find any trace of the 1st MO Eng. (or Wm R. Barnum of the 23rd Iowa Inf).
I am in the midst of transcribing the Roll of Honor for Jefferson Barracks into a spreadsheet so I can search different ways, but as it is a very low priority item, it will be a long time before I get it complete.
A quick look indicates that there were several UNKNOWN graves created at the same time that the 1st Iowa Cav soldiers were reinterred which might suggest that the members of the 1st MO Eng were not identified. As all of the personal papers and effects were taken from the soldiers, there was not much to identify them. As there were other members of the 1st Iowa Cav in the area, they might have identified their fallen brethren.
The UNKNOWN graves at Jefferson Barracks that originally came from Mexico just have the death date of Sept 27, 1864. Do you know if any of the slain men was a Lieutenant?
That is about all I know at this time, but if I come across anything more specific, I'll let you know.
Craig