If the LDB were interested in obtaining the throne, something like the Jagged Crown or being supported by members of the Moot would do far more to give them legitimacy. It might link them to old Nordic heroes and kings of the past, but it's not like the Amulet of Kings, simply possessing the title does not bestow legitimacy or a right to rule. Most likely though, no, the title of Ysmir would not give the LDB any legitimacy to rule Skyrim. One of them was indeed as a title given to Nordic kings, but it has also been the name of a god and is given to particularly important or powerful individuals such as Pelinal, the LDB, Wulfharth, and Hjalti Early-Beard (the latter two of which were given the title before they became rulers for reasons not necessarily related to the title of Ysmir and any legitimacy it may or may not give them).

The title Ysmir has had many different functions and meanings over time.