Note: The following argument contains spoilers from about halfway into Fullmetal Alchemist, and close to the end of Problem Sleuth. You have been warned.
Okay. This is the big one. Bolded for effect. I love Problem Sleuth. Okay? I archive binged the entire story in ONE DAY. Nonstop, I started that bitch and rode it all the way home. Do I regret it? No, of course I don't, because it was awesome! That is why is pains me so dearly to say... Problem Sleuth can't win this fight. Okay? He just can't. Guns are a nice thought, but what a lot of people don't realize is that Fullmetal Alchemist takes place in an alternate WWII of sorts. As hard as it is to believe, the technology Mustang has faced-and decimated, mind you- is actually MORE advanced than what Problem Sleuth has at his disposal.
And yes, Problem Sleuth has some special abilities, but the problem is most of those require one thing he simply does not have access to here: Allies.
Pickle Inspector, Hysterical Dame, Ace Dick, Fiesta Ace Dick, Zombie Ace Dick, hell even Snoop Dogg were all essential in Problem Sleuth's victory. But they're not here, and due to the nature of this battle they will not arrive through some contrived means. Without them (and without a portal to the imaginary world, if this is a truly neutral ground) then Problem Sleuth is simply an ordinary, if hard-boiled man with a gun, against Colonel Mustang, who can and has wiped out hundreds of soldiers with roughly the same equipment. And that was when he was a rookie.
Mustang has proven that bullets, while effective, have two problems when against him: He's tough to hit, and if he is hit, there's a good chance he'll kill you before you can gloat. He's stubborn to a fault, and this shows up in battle, particularly his battle with Lust, where he was impaled with what should have been a fatal wound, not to mention his gloves were wet, and thus couldn't be used. What did he, lay there and die? No, he (capped for effect) CARVED A TRANSMUTATION CIRCLE INTO HIS HAND, THEN USED HIS CRIPPLED FRIEND'S LIGHTER TO MAKE FIRE AND SEAL HIS WOUNDS WITH FIRE, THEN WENT ON TO BURN LUST TO DEATH, WHICH TOOK OVER TWELVE SICKNASTY BLASTS OF ALL-CLEANSING FIRE.
I may be ranting. In fact, I'm fairly sure I am. But outside of Problem Sleuth's own world, he is little more than a Badass Normal/Cloud Cuckoolander.
Plus, chances are his gun will turn into a key when he tries to fire. That sort of insanity backfires against most enemies. I'll think of more to add later, until then please take my points into account. Remember, this is not a popuarity contest, but a test of combat.