LS has A Hero's Resolve objective in play adding one force icon to each card in its edge stack. DS plays Reversal of Fate switching the edge stacks. LS won the initial edge battle 7 (5+2) to 4. Does the LS's card ability transfer to the new edge stack? If so, it would still win the edge battle 6 (4+2) to 5, correct?

A Hero's Resolve is a constant ability that adds 1 force icon to each card in the controller's edge stack. Reversal of Fate switches the edge stacks (before they are revealed). End result: the LS player now controls A Hero's Resolve and the edge stack with the DS cards and that is the edge stack that Hero's Resolve adds to.

So, looking at the text...

A Hero's Resolve

While this objective is undamaged, each card in your edge stack gains 1 Force icon when resolving an edge battle.

Reversal of Fate

Interrupt: When both players are done placing edge cards, and before edge stacks are revealed, switch edge stacks with your opponent for the resolution of this edge battle.

So in the example you gave, Reversal of Fate actually couldn't be played - the edge stacks were already revealed, so the timing for Reversal of Fate had passed.

Assuming Reversal of Fate had been played at the proper window, then yes, the effect of A Hero's Resolve would apply at the resolution of the edge battle, after Reversal of Fate had taken effect, and the Light Side player would win the edge.