I have no idea why I tried to do sub 10 minute attempts on this. My first real attempts yielded an 11:08. At that time I had some ideas for new strats that I figured could bring it sub 11 pretty easily.
My next batch of attempts yielded a 10:43 or so, which I thought was pretty good at the time. Then I somehow got a 10:17 shortly thereafter and had no idea how that was possible. That was quickly followed by a 10:12.5. Another attempt later on that night was 24 second ahead of that at one point, then Drac and Drac 2 ate all of that time through pure RNG.
I did some off-stream attempts a few days later and brought it down to 10:02.5. At this point, any further improvements could only come from boss pattern RNG (Technically that was also the case when I got 10:12). Being only 2.6 seconds off of a sub 10, I figured it was only a matter of time before I managed sub 10.
I'll say this: Grinding attempts for purely RNG based time gains is not really that fun. I could have easily submitted the 10:02 and it would have looked fine. But I wanted to beat that arbitrary number, and I'm pretty sure my sanity hated me for it.

Either way, I crushed the 10 minute mark when I had a decent early fights, putting me about 4 seconds ahead going in to Minotaurus, lost all that and then some up until Dullahan, then gained it all back between Death, Legion/Nuculais, and Drac 1. Drac 2 gave me almost perfect RNG, netting me about 13 seconds.

Actual run comments:

Crazy Armor:
Got an unlucky pattern that cost about 0.5 seconds thanks to getting hit in phase 1. Otherwise perfect.

Wyvern:
Only way this could have been faster is if I could have gotten a couple extra hits in before he took off. This would have allowed me to use one less Windmill attack, and saved up to a second.

Trevor:
I usually lose a bit of time here, so almost breaking even is really lucky. Trevor has a guard counter that decreases every time I hit him with the whip. When this counter hits 0 he'll start guarding any incoming attacks. The counter seems to increase over time when he's not taking hits. The Knife Crush works well to start with, but if used during the latter half of the fight, Trevor will block almost everything and prolong the fight.

Skeleton Diver:
Got a really good start, bringing him down about 30% before he submerged. Fish did his laser after the Diver jumped on the platform, which is the quicker of the 2 attacks it can do while on the platform. Diver jumped back on the fish almost immediately after the fish submerged. Managed a lot of double hits when the Rider was mounted. Overall a really good fight.

Minotaurus:
Did the strat perfectly, but the 2 jumps at the end cost about 4 seconds. I also screwed up the perfect guard on the first jump. Time loss here was pretty much all RNG.

Isaac:
OK fight, didn't get the best luck, but not the worst by a longshot. Isaac has a chance whenever he's about to get hit to either dodge or block, and there seems to be no limit on how many times he can do this. It's possible to lose up to 15 seconds in this fight purely from RNG. His ID, however, cooperated until the very end.

Saint Germain:
Got an unlucky teleport at the very start of the fight, and he never appeared where the auto-target aimed at. Lost a bit of time here because of it. When Saint Germain drops below 50% HP, his next move will always be Time Advance, but this can be overwritten by using the Stopwatch since his AI will always counter Stopwatch with Time Stop, and this is always #1 priority.

Trevor:
When his white aura is present, Trevor cannot be made to flinch. Thanks to the Holy Whip, I can knock him out of the aura in 3 hits. Execution-wise, this went perfectly, but I could have gotten slightly better luck with how many times I was able to hit him each round before he started blocking. He has the same block mechanic he uses in the first fight.

Dullahan:
Lost a bit of time here due to bad RNG. He ran off early, and that beam attack is hard to avoid while keeping up damage as Trevor, since the Windmill attack takes too long to get off in between rotations, but only 3 whips is too quick to quick step through the beam. I lost time from taking the hit and flying into the wall, but it could have been a lot worse if I had gotten hit from a different angle. Ultimately I want him to do his other charge (where he points the sword straight down) since he takes double damage during the charge up for that.

Isaac:
Isaac has the same block mechanics he had in the first fight, which makes actually connecting with whips random. He lacks the dodge mechanic thankfully. I need the knives always on him to keep him from summoning any IDs. He didn't block enough to force me to use a 3rd Knife Crush, so that at least makes this fight average. I use the restoration circle after this fight since it's the fastest due to being able to be on top of it when it appears.

Death:
This is where I started gaining time. Death gave me pretty much the perfect pattern by only doing Scythe swings. I use Grand Cross to avoid it and keep the damage up. Once he drops below 50% his next move will always be Fireball, which with my maxed out attack power enables me to kill him before the fireball expands enough to hit me.

Legion:
I let the Knife Crush do damage as I'm climbing the staircase, then switch to Axes when I'm level with Legion. Axes do 90% of my attack power per hit (this is dependant on what whip I have equipped, which I why I stick with Square Tip instead of switching to Legion's weakness of Holy). The Axes with their superior range allow me to whittle Legion's HP down in no time flat. Note that Legion cannot die until the core starts glowing purple, which only happens after all the exterior pieces have fallen off. Using Axes also allows me to stay at this level instead of dropping back down to the ground, which lets the Knife Crush hit him a lot sooner than if I were to jump out and whip him to death.

Nuculais:
Gave me pretty much the perfect pattern. The rush attack he did actually helped when it hit me since it knocked me back to where he was standing, allowing me to get him down past 50% before he teleported away.

Dracula:
Meteor is the most preferable attack since I can get a full combo in so long as I attack him from the back. He did one blood suck as well, but this was an extremely lucky fight since he did 2 Meteors.

Dracula (Demon):
If he does one of his mega attacks, it wastes up to 8 seconds. Thankfully he immediately came to the platform. Got a pretty lucky pattern in terms of what attacks he used. Claw Swipe is the most preferable since it's the easiest and quickest to avoid/block. Overhead Smash is take the longest to dodge, but can be perfect guarded somewhat easily. Straight Punch and Backhand are almost impossible to perfect guard due to the random length of his windup, and how fast the actual animation is. He also laughed 3 times, which is just free damage. Getting a time this low requires that he be killed before he moves away from the platform, which is completely random. If he moves away, it's a 10-20 second loss depending on how many attacks he chooses to do.

Closing: I have no intent to improve on this. Any further improvements can pretty much only come from RNG, which I was already fighting pretty hard. I beat my goal time, and am quite happy with that. I wish anyone who attempts to beat this the best of luck, because you'll need it when battling an RNG this stacked against you.

Thanks to my stream viewers who stuck around and actually watched me grind attempts on this. I know watching resets due to bad RNG isn't the most interesting thing to watch.
Also want to thank the whole crew at SDA for hosting all of my other runs, this one, and hopefully more in the future.