Based on what /u/Elardi said, I am going to be changing this from typical FFA rules (which is what Elardi uses) to Duel rules. Note: This assumes good MP experience and having played Duels before for both Batman and Lex.
This gets rid of a ton of starting bias factors. I'll take away the norm of mirror matches.
There are a few Civs that have incredible early game advantages, and Duels rarely make it past Crossbow era because of how much better Liberty is early game.
Who would pick who? It's a tough choice.
Lex - The Huns
Lex is interesting because his arrogance and aggressiveness changes based on the story. Based on my limited knowledge of him, I believe he would choose the Huns based on the overwhelming force they provide. Horse Archers and, to a lesser extent, Battering Rams can overwhelm an opponent very quickly with no real sacrifice.
Horse Archers require not strategic resource. They are very early in the tech tree, and they're cheap production wise. You can plop 2 cities down and then pump Horse Archers the entire game and most likely win. It's very difficult to stop a competent player with Horse Archers.
However, I have seen it done. Babayetu, a ridiculously good MP player, has gone up against other equally skilled players using Horse Archers and has won. It is possible, but it takes good land and good use of that land.
Batman - Russia
If Batman was smart, he would also choose the Huns. But... that's not interesting really. So I'm going to give him a different Civ.
To combat Horse Archers, I noted that we needed better land. We need a clear advantage over Lex in order to win. To that end, I choose Russia.
Russia's unique ability is that it gives you double the amount of strategic resources on a tile, and also +1 production on each of those resources. This gives Russia a huge production lead early in the game, which is exactly what we need to do in order to counter Horse Archers for long enough to out-tech them and make them obsolete (by getting to Crossbows).
Additionally, the double Horses each horse tile will receive means that Russia can utilize enough Chariot Archers to even the odds early game. Although they are at a disadvantage, there will be enough of them to make a difference.
The Game
Starting location, terrain, and natural wonders are the biggest random factors in a game like this, since we've evened the playing field in all other ways. Now it just comes down to skill.
City placement will be huge, and Batman needs to set up easily defensible blocker cities in order to create a choke that Lex can't easily push through. If Batman is not able to manage this, he will lose from the tides of Horse Archers plowing through his lands. These 2 cities need to be nestled in Mountains with Rivers cutting off the movement of his Horse Archers. This relies on good Terrain being around, which, again, is random.
In the end, I believe Lex will win this one overall. Batman needs to rely heavily on the use of good blocker citites which Lex doesn't need to. Lex needs to win before his GPT is so low from his massive army he's not making any science.
As soon as Lex starts seeing the "No Science" on his screen and Batman is still positive GPT, Batman has won.