I've tested it a bit, it works with Charge, Heart of Fury, and Vengeful Defeat, at least.
My hypothesis was that melee+ranged dual wield full attacks would work for Full Attack abilities that have their own animation (as opposed to using the melee swing animation), but I also tried it with Whirling Slashes and it doesn't work there.
As for Hand Mortar, I quickly checked a couple things, but it gets slightly complicated.
With Heart of Fury, there is no problem. Every hit triggers Hand Mortar's AoE, Blinding Smoke, or Powder Burns. It also triggers Kitchen's Stove Wild Barrage's bounces, and Eccea's Arcane Blaster's modal. However, on Charge and Cleave, I only seems to get the Blinding Smoke, and not the AoE damage (Although they do trigger Eccea's Blaster's modal, haven't tested Kitchen Stove yet).
I doubt it matters but my test were done with melee in main hand, ranged in off-hand.