Not that it matters since any biographic connections this story makes are irrelevant and distorted, but the actual Beethoven was gung-ho about piano duels, so he'd never refrain from entering a modern piano contest.
He needed to be: in order to become known in the cut-throat musical scene of late 1700s Vienna, and receive the patronage of musically-inclined nobles, he had to prove himself against rivals in performances. He did, too: he was considered peerless in piano duelling, especially when it came to improvising on themes given on the spot, and usually wiped the floor with his opponents.
But I guess this behaviour wouldn't go down well in a fluffy story like this.