Initially, I was shocked that he knew the names of more than one classical composer and secondly, I was shocked that I had never considered this myself. The question, while slightly inane, does bring a certain humanity to historic figures that are often set upon pedestals as gods of composition. These high and lofty figures were mere mortals in their day with strengths and weaknesses. Besides, why shouldn’t we pit them against each other for our own amusement?
STATS
Wolfgang Amadeus Mozart

Born: January 27, 1756
Lived In: Vienna, Austria
Composed: Nearly 1,000 works in just over 30 years including everything from simple piano songs to epic operas. His most famous works include The Magic Flute, A minor Piano Sonata, and his Requiem.
Fun Fact: He was a very fashionable guy who always had the best clothes and wigs money could buy.
Ludwig van Beethoven

Born: December 16, 1770
Lived: Bonn, Germany
Composed: Just over 200 published pieces exist from his lifetime as well as dozen of unpublished sketches. His most famous works include Piano Concerto No.5 (Emperor), Symphony No.5, and the massiveSymphony N0. 9.
Fun Fact: He started to loose his hearing at the age of 26.
When setting up this match, important things must be considered. Location for one. I would hold the fight in Vienna. Mozart did travel all over Europe as a child prodigy but his favorite place was Vienna. Likewise, Beethoven might have considered Bonn his home, but he did spend time in Vienna in his 20′s studying under the top composers and music theorists of the day. Vienna would be the most neutral territory for the two.
Another consideration must be the referee. We would need to have an individual that while respecting each composer’s talent did not have a definite bias one way or the other. I would suggest bringing Hayden out of retirement to judge the match. Hayden was a teacher to both and saw great potential in both composers.
Now, for the Battle Royale.
‘The bell rings and the composers approach each other. Mozart, being the excitable little scrapper he is, throws the first punch using his impeccable counterpoint he mastered while still in puberty. Beethoven is stumbles back, but this is nothing he hasn’t seen before. He counters with a one-two punch using his ability to develop a theme and genius use of codas. Mozart is shaken by this since it in no way follows the musical forms he himself had mastered. He quickly retaliates with his innovative comic operas but it’s deflected by the strength of Beethoven’s symphonies. 1, 2, 3, 4…9! 9 punches right to the throat (He wrote 9 symphonies). Beethoven thinks he’s won but while his back is turned, Mozart takes he out with the sheer prolific volume of his compositions. Beethoven is down for the count. As Hayden is counting, Beethoven struggles to get up. I don’t believe this! He’s standing again! Mozart looks nervous. He’s got nothing left to throw at his opponent. Beethoven throws down Mozart using his pent up Daddy issues. That’s right folks, he had to put his career on pause in his late 20′s to take care of his family because his father was a belligerent alcoholic. There’s a lot of pain there folks.

I guess it doesn’t matter how much music you write, when it comes to a fight, he with the most issues wins and Beethoven had issues. I won’t even get into his “Immortal Beloved.”