I finally managed to conquer those Castle Crashing "The Beard" badges- so here's my video of how to do it.
1: As the level starts, stand on the middle left platform dodging his attacks. After the second attack, head to the platform to the left, and continue to hit him until he does hit "lazer eyes" attack. If you need to move around to prevent getting hit, do so. After he does the first lazer attack, get into a position so that he can't hit you with the second attack, and strike him as many times as you can.
2: Repeat this strategy, but on the right side of the map.
3: Repeat steps 1 & 2 until he turns into a skull.
4: When he's a skull, head toward the left-most platform. Wait for him to head towards you, and once he gets within your attacking range, attack him as much as you can. When he drifts away to do his fireball attack, head to the rightmost platform.
5: Do the same thing as step 4, only on the right side of the map.
6: Repeat steps 4 & 5 until you defeat him.
As you can see, I took about 3 minutes to defeat him- which is good considering that you need to do it in under 4 minutes for the hard badge. If you are able to combo-up your attacks just right, you may be about to do it in under 2 minutes.