In addition to TheWingedAziel's recommendation of fighting The Chain for this, I'd heavily recommend fighting The Edge. The Chain may be significantly easier, but it's way quicker to fight The Edge and there won't be any concern with having enough time. You can beat The Edge in less than 3 minutes if you've practiced him enough and don't get hit. I actually got this on my first playthrough by dumb luck (it was my 5th or 6th try at the fight but still only the first playthrough).
So here's a few quick tips for beating The Edge quickly:
Obviously, play on Furi difficulty, not Furier.
For the first two stages of the fight, there's a really easy way to get your hits in fast. As soon as you parry one hit from The Edge, throw out a normal attack of your own. It'll knock him back and then he'll try to attack you again. There's only two instances where this won't work: the first is that The Edge will do some area-of-effect slices (marked floor attacks), in which case you have to dodge them until he reveals an opening, and the second is that when you counterattack him, he will rarely block your strike and come at you with another of his own. In that case, just parry him again and counterattack again. From my experiences, you'll get your hit on the second try no matter what. Only throw out one strike at a time though, because he will dodge your second.
The best way to know when to counter The Edge is by relying heavily on the audio cues. He likes to mix up his attack pattern and rhythm a lot so if you stay in the same rhythm every time he starts attacking, chances are you're going to prematurely parry a lot and get hit for it. Listen for the sound indicator that happens along with the flash indicator and parry when you hear it, but not before.
On the last stage of the fight, The Edge will be able to take two hits at a time instead of just one if you're quick enough. Also, the best strategy that I've found for dodging his lengthy fire projectile strings is to dodge forward past one, then dodge backwards immediately. Rinse and repeat until they've all passed you, then get your one or two hits in.
Using this strategy, I can get past the first stage of the fight within 15 seconds consistently, the second stage within 45 seconds to a minute, and then have the fight done with by the time the clock hits 2:30.