For what it is worth: in 1864 the veterans of the 25th Missouri and the Engineer Regiment of the West were merged into the 1st Missouri Engineers. The 25th had long been acting as engineers, working on fortifications at Corinth MS, New Madrid, and Patterson MO. The Engineer Regiment, originally Bissell's Engineers of the West, had been engineers from the summer of 1861. The combined 1st Missouri eventually accompanied Sherman on his march to the sea and through the Carolina's.
The 25th was originally to be organized as the 13th, but they were captured at Lexington and when they got paroled found that unit designation had been taken so settled for the 25th. Interestingly, the 25th fired the first Union shots at Shiloh having been quietly sent to find out whose forces were out beyond the Union camps.

The history of the Engineers is frequently confused.