One thing that may be true for him is this; he may have been shipped home with another unit. This happened many times at the end of the war.
The 360th was an independent engineer unit, so were never actually part of a division. These were called "bastard" units. My father's unit (the 540th), was an independent regiment, too. They were often attached to other units, such as divisions, regiments, companies, etc, but were always under the control of an ARMY or CORPS. So in other words, they were assigned where needed.
Regarding the fire in 1973 in St Louis...This was not the only place where personnel units were kept. More often than not, you can still get copies of the DD214's from other sources, including the Veteran's Administration. If you click on that link, it will take you to their site. Then simply click on the related state for your step-brother. You will be provided with contact info. This is how I obtained my father's discharge papers. Other venues include his local county building or the local funeral home. Hope this helps.
Proud Daughter of Walter (Monday) Poniedzialek
540th Engineer Combat Regiment, 2833rd Bn, H&S Co, 4th Platoon
There's "No Bridge Too Far"