Provoke is off-GCD; it shouldn't take you a GCD's worth of time to voke spiny and get back to whichever sister you take.

First off, for first sisters (no spiny) the most optimal thing you can do is Fast Blade when Garuda summons sisters (DIEDIEDIE), immediately Savage>Halone Chirada, start running.

For the sisters after Aerial Blast, you have to tackle it a little differently but it's the same idea:
1) After spiny dies, Garuda comes bask for Mistral whatever. Use Fast on her after the cast bar finishes.
2) Garuda teleports, does the feather rain thing. You should run away for this and then run back in when the aoe marks fade.
3) Savage Garuda when she returns, then Halone your sister of choice. Combo should last long enough. If not, do same as first example: Fast Garuda, Savage add, Halone add.

You can honestly leave spiny on whoever it is on during this phase for 2 stacks and take it back after the sister you are holding dies, unless your DPS is slow/bad. Then you have to take it back early but your hate should be fine.

Should not have hate troubles doing this unless your DPS out-ilvl you(r weapon) by 20+.