How can I take over when the Main Tank dies?

I've started off-tanking 60 ex primals on my Dark Knight lately, and the majority of the fight I spend outside of Grit to avoid the 20% damage penalty when there are no mechanics for me to deal with. However, sometimes the tank dies to a late heal or a missed cooldown, and I'm scrambling to pick up the boss.

I can usually get off a Provoke immediately, but then I'm left with a problem. Most of the dps are in the middle of high damage rotations or the healers have regeneration mechanics up, and so they rip threat right back before I can establish an enmity lead.

Between Grit, the 2.5 second GCD, and the Power Slash combo, it can easily take 5-10 seconds to solidify my position at the top of the target's list. That's often plenty of time for a boss to cleave the raid and make the situation worse.

A coordinated party might be able to hold back for a bit, but that doesn't help with healer enmity from pre-established effects, and is difficult to pull off in a Duty Finder group.

How can I quickly take over in a bad spot as off-tank? Is there something I'm missing?