Don't change the character's name, at least not in the narration where the character is thinking of himself as the new name. Don't start referring to a male-turned-female character as 'she' in the narration, for exactly the same reasons. Unless you are a Jack L. Chalker character (in which case I will upon your request happily shoot you to put you out of your genderbent, bodyswapped, identity-lost, species-tossed misery) you are not going to start thinking of yourself as someone of a different name, gender, or species, without several years of extended emotional "WTF? OMG, no, seriously WTF, fix me!one!1!" (Even if that's presented in a comedic manner instead of an angsty one.)
And once that happens even in a natural time-progression it's going to be very hard to keep me as a fanfic reader who gives a damn about Spike, Xander, Angel, etc. interested in this to-all-extents-and-purposes original character, without a really good re-creation of the tv character's mental and emotional makeup. The more and faster they adapt to changes to their very identity, the more likely the story is to lose me. It's hard enough to re-create/hold onto recogniseable TV characterisation when we have a name and a physical description to hang the recognition on -- there are plenty of Spikes, Xanders, and Angels in fanfic who we tilt our heads at and try, really try, for a while, to see our guys in, before finally blinking in confusion and giving up.
Giving them a new name and gender isn't going to make it easier for readers to buy changes to their personalities (though I can follow the possible authorial logic behind thinking it might); it actually requires a better grasp of the inner character, to carry us through all the changes going on. And while it's possible to show us the character changing internally as well as externally -- in fact, it would be unrealistic if he/she didn't -- there's a reader-loss danger in carrying that change too far, even if it's realistic to the plot. There's a reason I'm ragging on Jack Chalker above; anyone who's made it through Book 4 or 5 of the Dancing Gods or Soul Rider series will take my point, I think. Readers who like original characters or massive challenges to character identity will probably stick with the story. Me... I will be frowning sadly, because I like the concept of physical change and genderswap stories, but it's so hard to find one where I can continue caring about the characters.
1 Anyplace you see "we" or "readers" please read "me and anybody who happens to agree with me." Not actually speaking for anyone besides myself, just finding it hard not to write in the plural for some reason.