We do have to consider the distinct possiblity that Lincoln may pull an Oddjob and decapitate Paul at 50 yards by throwing a blade-rimmed tophat at Paul's neck. While I would like to just say "Lincoln wins again!" and walk away, there is actually a lot of interesting physics in this scenario we must explore.
Take the FUN BUX that Paul keeps in his coat pocket, for instance. What we know about Silver is that it is dense and semi-malleable. This is very important from a kinetic perspective, because it means you can stuff a lot of mass in a small area and it will deform on impact.
When you throw an object, the kinetic energy it caries is proportional to the mass and the velocity squared multiplied (mv^2). If you want to throw something with a lot of energy you need to throw it fast, as it will have exponentially more energy with a higher speed.
Obviously this means you want something light to make it easier to throw fast, but if the object is too light relative to the mass of the target, nothing will happen. This is why bugs go splat on the fender without damaging the car even though the speed differential between the bug and the fender is ~70mph. The bug is simply so light that it cannot carry enough energy to make a dent in the car.
So if Paul chose to throw a ping pong ball, which is light enough to throw fast but also small enough to fit in a pocket, he still couldn't deflect the blade-rimmed tophat flying towards him. It's so light it would bounce off without appreciably affecting the trajectory.
But a silver FUN BUX coin has enough mass relative to the tophat and is small and easily throwable that it fits the bill for deflection perfectly even if it is useless for 'fitting' (i.e. paying) actual bills. It also has the bonus attribute of being slighty malleable. This means that if Paul aims for the blade-rim and impacts it, there is a chance that the silver could actually deform on impact.
Who cares you say? Well what this means is that even more of the kinetic energy will be transferred to the tophat before the coin bounces off. You will find that this exactly why cars are designed with bumpers and hoods that deform and crumple under impact to absorb energy. In those cases, it's to safely decelerate a crashing car. In the FUN BUX example, it is to get maximum deflection from what may be a sub-optimal impact trajectory.
You can't expect Paul to throw perfect deflections every time, after all.
So in this exchange, we have a draw. Ron can stop the decapitation, but his FUN BUX are not massive enough nor can he throw them fast enough to use them as weapons.
But our brilliant champion Lincoln has prepared for this. You see, he keeps copies of the Emancipation Proclamation in his coat pocket. After having missed his window for an Oddjob style knock-off, he will simply pull out the Proclamation and drop it on the ground.
The sight of such an evil, 'right destroying' document will send Paul into a primal rage. He will not be able to stop himself from charging at the document like a bull toward a matador's cape. So he charges at it and Lincoln side steps like a master player. Now Paul is lined up perfectly as he is bent over, tearing up the Proclamation on the ground with his teeth and pounding his fists into the ground while screaming "TSA RAPIST!!!!".
This is the moment that Lincoln brings the ax down on his neck. Licoln wins
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Now this is interesting. I think it's safe to assume that Paul would actually bring a battle flag to the brawl. I mean, if he's gonna take down the worst dictator of all time, he might as well make a statement about lolState's Rights while he's at it.
Being crazy like a fox, we can expect that he'd also tip the flagpole with a spear point and expect to use the flag itself as a strangulation device in a pinch.
However, due to material shortages caused by the war, the thread count of most CSA flags was rather low (most unlike ACW CFC threads, btw). So if he tried to choke Lincoln out with it, it would most likely rip on his awesome manbeard.
So let's say he went in for a spear thrust. Ok, game over, right? Not quite.
You see, a little known fact is that Lincoln wore a tophat both for decapitation purposes and to conceal the American flag bandana he kept tied around his head pirate-style. While you might expect that Lincoln would be a little more conspicuous with his USA#1! apparel, you have to realize he was the O.G. and as such didn't need to flaunt his pride. Everyone knows he's got it.
Now due to Northern rejection of slaver cotton, this bandana was made out of fine Chinese silk with superb threadcount. It's strong and more than capable of a chokeout.
So Paul goes in for the stab and Lincoln front-flips over his head while untying his bandana mid-air. He lands behind Paul and in one precise, well-practiced motion, slips the bandana over his head, around his neck and cuts off circulation.
Lincon wins, USA#1! style.