Using the key, "Aslan is Christ," I am even less satisfied now with the implied theology behind Aslan's sacrifice for Edmund than I used to be.
If Aslan were intended to be merely a Christ-figure, it would be just what I think a lot of people take it to be: a fairy tale parallel to Christ's own sacrifice, with Edmund as fallen mankind.
As it is, though, hasn't Christ already died to save Edmund? What does it say about the Cross at Golgotha that it wasn't sufficient to save a particular human being?
In fact, Aslan's death saves, not Edmund's soul, but his natural life. I've already pointed out that Aslan's presence in Narnia has essentially no Trinitarian dimension, that his presence as an incarnate lion has no evident purpose. Now we have Aslan dying –- in a passage some Christians refer to with great reverence, almost as though it were truly about Christ -– merely to save a single life. Does Christ in the Gospel preach so great a concern for the death of the body?
Aslan's sacrifice on the Stone Table may be one of love ("no greater love hath a man than to lay down his life for his friends"), but it is an oddly narrow love for God to have, since there are uncounted numbers of creatures Aslan allows to die (including, a few years later, Edmund himself).
More than love, actually, Aslan's death has the character of utility. It's a means of destroying the Stone Table, which artistically is something of a gimmick anyway, and overthrowing the White Witch by fooling her into a bad bargain. And since both the trickiness of the bargain and the need to overthrow the White Witch were built into the Divine plan for Narnia, the whole thing seems awfully contrived.
Lewis's concern was to write a set of children's stories that suggest elements of the Christian faith, not to develop a rigorous and complete Aslanian faith perfectly consistent with Christianity. Hence Aslan offering his life for a traitor, then coming back to life and overthrowing a principle of evil. Hence too the scene at the end of The Silver Chair, in which we learn it is necessary for a thorn to be driven into Aslan's paw to restore King Caspian to life. We never learn why it is necessary, nor whether Aslan does this for each saved Narnian, nor is it clear whether Caspian is resurrected body and soul. It is enough for Lewis's purpose to suggest that, somehow, Aslan's blood painfully drawn (pain? in Aslan’s country?) brings eternal life. But that is not enough for the implied theology of the Chronicles of Narnia to either be wholly consistent with or to always meaningfully reflect Christian theology.