If Rand does as you outlined, I guess he could get by, most of the time, looking as he currently does. And I suppose if he ever did have to go out in public where he might be recognized, he could always use a Mask of Mirrors disguise.

But, I have another reason for thinking Rand will look different.

In your theory (I just finished reading it), you mentiond Egwene's dream of a "man lay dying in a narrow bed, and it was important he not die, yet outside a funeral pyre was being built, and voices raised songs of joy and sadness". My take on that dream is, Rand's old/dead body is on the funeral pyre (as you suggested), and the "man who lay dying" is Rand in his new incarnation (having just been yanked from TAR looking different, or having just somehow survived in Moridin's body). And the fact that he looks different explains why Egwene doesn't recognize the dying man as Rand. As for why the new Rand lays dying ... either it's just after he was ripped from TAR, like Birgitte was dying before she was bonded, or he's only barely surviving in Moridin's body, and Nynaeve has some work to do to save him.

Also, the other dream of Egwene's you mentioned - "Rand wearing many masks, until one of the masks is no longer a mask, but him", is easily explained if Rand's final incarnation(mask) looks different from his original appearance.