U.S. copyright law has historically identified 3 Performing Rights Organizations ( SESAC, ASCAP and BMI by name in Section 101 of the Copyright Law. To the extent rights To R’ Shlomo’s music have been assigned, one of these entities would be the centralized source for rights (and can be confirmed through a simple online query.
P.S. The OP asked about music rights, not about R’ Shlomos well-known hashkafah and personal issues. Many frum shuls still use his nigunim nothwithstanding these concerns., some real and some questionable.