I have been researching my uncle, Chester Higdon, who died at Saipan. He served in the 165th Regiment of the 27th Division, US Army. I have collected some Army reference books in my research. Please understand that I am not an expert. However, in researching your grandfather, you must lay out all the information that you have, factual as well as hearsay. They may provide important clues in obtaining the information that you seek.
I found this excerpt from U. S. Army in World War II, The War in the Pacific, Campaign in the Marianas by Philip A. Crowl, p. 124: "Each of the three Army regiments had its own shore party battalion - the 152nd Engineers for the 165th Infantry, the 34th Engineers for the 105th, and the 1341st Engineers of the 1165th Engineer Group for the 106th Infantry."
I am sure there were more Engineer units in this campaign. I just have not researched the entire order of battle for the Army in their invasion of Saipan. Once you find out your grandfather's unit, it should be rather easy to trace the unit to see what combat they saw.
If you want to exchange more information via email, feel free to do so.