Hey, this isn't a bad idea.
Okay, I'll try to do option #2.
As with most Marvel/DC crossovers, the two heroes fighting is usually the result of a misunderstanding or something or another. The story itself would have Logan thrown into the DC Universe thanks in part to him stopping some nefarious plot of Apoc's along with his fellow X-Men. Apoc builds some machine thingy-majig that would enable him to breach universes, but Logan slashes the core drive with his claws, causing him to be warped, atom by atom, into the DCU by his lonesome.
After getting readjusted to a world that doesn't hate and despise him purely on his being different, Logan decides to chill and hit the nearest bar which just happens to be in Gotham City. While there, he meets Matches Malone, who is busy antagonizing some of the local ruffians. Eventually a bar fight breaks out, and Logan ends up wrecking the opposition, being pushed to the point where he pops his claws. Before he can permanently rearrange someone's face, Malone stops him. Not appreciating the guy responsible for starting the whole fiasco trying to prevent him from asserting himself, Logan backhands the holy hell out of Malone, sending him crashing to the ground. Fed up, Logan decides to leave, but not before briefly smelling Malone's scent tailing him and then vanishing altogether.
Malone, obviously one of Batman's aliases, returns to the Cave, curious on the newcomer to the Gotham underground scene. Naturally, he turns up nothing on the Bat Computer that would give him any relevant info. He does, however, have a hair fiber he had plucked from Logan and decides to do some more in depth research. He discovers that the newcomer has a DNA sequence unlike anything he has quite seen before, deducing the man is no doubt a metahuman of some sorts. Concerned a man such as he could possibly cause more problems in his city, Batman decides to go on a hunt for him.
Eventually, after some trying, Batman locates Logan, who is about to cut Killer Croc to bits (my homage to Marvel vs. DC
). Batman fires a cable around Logan's wrist and tells him to cease and desist, allowing the injured Croc to escape and lick his wounds. Logan, recalling the scent from before, and pissed he's being stalked, attacks Batman. Batman is immeadiately thrown on the defensive, surprised at the speed and ferocity of Logan. He also notes that despite the frenzied appearance of his attacks, the wildman is most definetely skilled and not just attacking randomly. Logan likewises concedes the "flying rat" knows what he's doing, but also realizes the guy is tiring ever so slightly where as he is not.
Eventually, Logan manages to cut into into the fiber of the Batsuit, drawing blood. Realizing that he can't afford to dodge forever, Batman throws a volley of batarangs at the man, attempting to buy sometime to regroup. Logan cuts them all but one out of mid air, but narrowly misses the final one which strikes him sqaure in the neck. Horrified, Batman attempts to offer aid to the man, but stops as Logan nonchantantly removes the bladed weapon and heals in front of his eyes. Batman, grimly realizing that he can't afford to engage Logan in prolonged combat, presses a button on his utility belt before hurling a smoke grenade at Logan, which has little to no effect.
A cat and mouse game ensues, with Logan's senses and tracking ability being a match for Batman's evasive ability and knowledge of Gotham's rooftops. Finally, Logan corners Batman and begins to demand answers, answers about where the hell he is, who the hell Bats is, and just why the hell he doesn't just fight him like a man. Batman smugly responds that directly, he can't best Logan at least not now that he's injured. He then tosses his utility belt at Logan, remarking that he has to "think outside the box". Logan's senses inform him of the smell of explosive gel, but the belt explodes with enough force to literally knock Logan off of his feet as well as cause significant surface damage to his skin and muscle layers. Batman pauses for a moment, waiting to see Logan's skin begin to heal before pressing his advantage, confident that he can't really kill the man. He employs a series of quick nerve strikes which serve to halt the healing factor, if only slightly, while managing to avoid a frenzied swipe from Logan in the process. He presses a button on his gauntlets, activating an electro-magnetic sort of "knuckle dusters", which fortifies his blows against Logan's metal skeleton. Eventually, Batman gets enough of a upper hand to cause Logan to tumble off the roof, but not before the former stabs Bats, once more in the upper shoulder. Logan's body crashes to the street below, as Batman, injured severely, makes it back to the Batmobile. He gathers himself and the KOed Logan and travels back to the cave.
Batman realizes that he only narrowly defeated Logan, and even then, only with his gadgets as he takes the mutant to the cave. Logan, who was only rendered koed for a moment and played possum to get to the cave remarks that Bats is a smart and cagey opponent with some "nice toys". I think it's a pretty fair representation of how Batman should beat Logan.
/end long post
