Mixed ink at the start is for SP regen to kickstart the Nebulous Duality active skill. It triggers Fyreheart and snowballs into enough SP for the rest of the fight. If you use much better equipment (which you absolutely should), you might find that you don't need AE as leader and can use LE as leader instead to bypass the opening SP regen sequence.
The battle opening is as follows:
1) BBB (all black ink) combo with AE
2) Nebulous active and switch to LE and use Mag active
3) Switch to FR for a DPS burst
4) Evade Tonatiuh air slam with FR
5) Switch back to AE and BBB
6) Switch LE, cast Mag, and back to FR for a mini burst
7) Switch to AE and BBB and hold onto it
8) Evade the air slam, consume the BBB combo, cast Nebulous, switch LE, cast Mag, and burst big with FR.
9) When Tonatiuh is regaining his 2 QUA charges, use FR ultimate and switch to AE
The skeleton rotation is as follows:
0) From (9) of opening, Tonatiuh will have just 1 QUA charge
1) BBB combo with AE, Nebulous active, switch LE and Mag active, and switch FR for DPS burst.
2) Switch back to AE and hold onto BBB
3) Evade the air slam and use AE ultimate, then consume BBB, Mag active with LE, and burst with FR.
4) When Tonatiuh brings back his charges, use FR ultimate and switch back to AE to repeat the cycle from (1).
It is important to have your FR's skills leveled up. And please, do not get greedy when Tonatiuh's hp falls really low. Note that his total hp pool is very small, at just 3 bars, so each length of the hp is much bulkier than it seems if you consider how long it takes to widdle him down. If he's not dead, add a rotation. You cannot go wrong with the conservative approach.