There is no apparent priority for issue of Sturmgewehr 44 to SS, rather to volksgrenadier divisions, and other infantry divisions. Panzer, panzer grenadier divisions, both Army and SS certainly had them, but not to the same extent as infantry, by proportion.
131st ID may have been first about March/April 1944. I have never seen any listed in formations in Normandy.
It is said that Hitler was not keen on them to start with, so they were issued to LSSAH Guard Battalion at Rastenburg, which won Hitler over.