I don't have a direct answer to your question. I don't fully know what constitutes the Messiah ben Joseph belief of the Jews since this is the first time I have read about it oo.
However, Biblical names are used in more than one context. For example Christ is called a "last Adam" in the NT. The term "Elias" while referring to an OT prophet also refers to others in the Bible. They are both personal names and used in a titular sense.
Anointed refers specifically to the Savior as one of his personal names but it can also refer to one who saves in a more general sense. As in Isaiah where King Cyrus a non-covenant person is referred to as Anointed. He is called that not as a personal name (or as the "Savior" Jesus) but as a title for one who saves (through letting the jews return and rebuild their temple) and as a type of Christ.
GUIDE TO THE SCRIPTURES - Anointed One
Jesus is called the Christ (a Greek word) or the Messiah (an Aramaic word). Both words mean “the anointed.” He is the one anointed of the Father to be the Father’s personal representative in all things pertaining to the salvation of mankind.
Isaiah 45: 1 Thus saith the LORD to his anointed, to Cyrus, whose right hand I have holden, to subdue nations before him; and I will loose the loins of kings, to open before him the two leaved gates; and the gates shall not be shut;
GUIDE TO THE SCRIPTURES Elias See also Elijah
There are several uses of the name or title Elias in the scriptures:
Elijah: Elias is the New Testament (Greek) form of Elijah (Hebrew), as in Matt. 17: 3-4, Luke 4: 25-26, and James 5: 17. In these instances, Elias was the ancient prophet Elijah whose ministry is recorded in 1 and 2 Kings. Forerunner: Elias is also a title for one who is a forerunner. For example, John the Baptist was an Elias because he was sent to prepare the way for Jesus (Matt. 17: 12-13)......
Thank you.