K, so that didn't go too badly. I actually lost time over your segments during the second to last and third to last segments. On Menace, I had an average first head, good second head, and terrible third head (got hit behind him a few times), but he didn't attack very often, so it probably went better than I thought, since it ended up edging out your fight time, which I thought was good. As patience consuming as it would be, his fight could go maybe 15s faster.
I ran out of MP for Erinys, but this was mainly because of missing. I think the number with which I missed could have accounted for the Bhuj damage I had to deal at the end.
On the third to last segments, where you start running through The Abyss, you can do a sliding dive kick to get past the fire pillars without taking damage. I don't think it costs any time (because you'd be in hit stun anyway) and saves 2-3 potions. Also, when dropping into the second room, where you SG to the lfet to avoid a fire pillar, if you SG to where you barely appear outside of the wall, the Rippers won't get in front of you 80% of the time, but will be close enough that they won't throw knives either, so you can do that whole stretch without damage. As another bonus, the Fire Demon doesn't appear either, so no worries on dodging him.
Also, Ninja Suit is not worth picking up in an in-bounds SG run. It adds 3 points of damage to the axe on Menace, which you only use on his first form, 44 -> 47. Basically, it doesn't make up the cost of going to get it by the increased damage output.
Lilith does seem to be worth it, 70 -> 73 damage on Erinys hits, of which there's more than 100, so 600 extra damage, and probably the only reason my fight time ended up faster.
You had mentioned (in your video description) that the fight might have gone better with another level up, but I don't think that would be very easy to do. From before Bat Company, I had a little bit more exp than you (I think it was just a couple hundred though, although I did level up a segment before you), and there's not a great way to get the extra level. I was 1751 from Lv. 20, and even with the couple rooms I have in mind, I don't know how to get that much exp. The best source would be the room of Harpies on the way to The Pinnacle after Bat Company, netting 480 exp. Other than that, it'd just be using Yorick on available monster that would die in one hit to it. Even if it boosts the damage output of Erinys by 1, that'd be 200 over the course of the fight and probably make up for the few extra seconds killing things would accrue.
So yeah, I think I'm going to work on a NG+ on another file, then come back and finish up the non-warp NG. Menace won't be nearly as much of a pain.