I know this conversation is old but I'm going to put my thoughts out anyway.
I would put my bets on Rand winning with ease. A few reasons for this. One the OP is inherently more flexible that anything the LR has.
There are a couple of things that so far I haven't seen considered. The first of these is angreal and sa'angreal. If we assume Rand doesn't have the Choaden Kal but does have Callandor then this fight is over in an instant. Rand unaided is able to wield at least 14 different flows, with Callandor how many do you think he could wield? Maybe 20 or more. Even with Atium the LR would probably still be obliterated in a second if Rand went all out and attacked with all of those flows.
The only thing I can think of to help the LR is being able to push on metal. The metal Rand wears would allow him to do so although I don't think the LR is strong enough to rip metal out of someone's body. He cold certainly push Rand about a bit but how long would he last against all those different attacks.
I would discounter the LR's emotionally manipulation for one reason. Once Rand has a hold of Saidin then his emotions are pretty much no existent thanks to the void. Compulsion and the like don't work on channellers when they hold the OP.
I would discount Rands taveren from playing a part as it's just too unpredictable and hasn't ever really seemed to help him in battle.
Overall I reckon Rand has this one. One massive wall of fire and it's all over. I don't think Rand wold escape uninjuried but certainly he would win.