I was originally going to title this "Is Elric the last incarnation of the Eternal Champion?" but decided to ask something more limited and maybe build towards that.
Some incarnations of the Eternal Champion, such as John Daker aka Erekosë, are able to remember other incarnations. This suggests that incarnations are in some sort of subjective sequence threaded through the multiverse.
Elric of Melniboné cannot remember any other incarnations (although there are episodes where he temporarily becomes a different one, and his companion Moonglum does remember past Companions). However, a unique thing happens to Elric at the end of the novel Stormbringer:
Elric, the entire world having dissolved into chaos around him, blows the Horn of Roland during the Conjunction of the Thousand Spheres, allowing the multiverse to begin again. Having done that, Elric's sword Stormbringer comes to life on its own and runs itself through Elric, sucking out his soul as it has done to so many others when Elric wielded it.
Michael Moorcock isn't precise about what the essence of the Eternal Champion is, but
Elric's soul, which Stormbringer devoured, has got to be a strong candidate.
In light of what happened at the end of Stormbringer, I could see where Elric might necessarily be the "last" incarnation, so there are no incarnations "after" Elric to remember being him.
Of course, evidence of a different aspect of the Eternal Champion who remembered being Elric would invalidate this theory.
Daker is the aspect who seems to remember the most past lives, but as far as I can tell he doesn't remember being Elric.