As we can see in the above picture, there are 4 'streams' of floating text (I turned off the purple 'DOT ticks and all party members' one because it's unreadable). We could have an option, then, which allows the player to have AST's cards be considered a separate number. The question is, where would it be best to place said text? I believe it best to put in the 'healing received' stream of text, along with 'Brutal Shell 3900', as a different color, customizeable in options (or turned off). Additionally, we don't need to have the text telling us the name of the effect, we could instead just have the buff icon to the side of the number. So it'd be more like...:
Fonts are enlarged for visual clarity in the example, they're smaller/not bolded in game. Customization options for the UI element added (the purple text here) would be color of the text, whether to show the icons or not (if not, just a regular number appears. Like an autoattack, with no damage typing symbol next to it), or to disable the values from appearing entirely. For ASTs, when the damage due to their cards is triggered, the effect would appear in their floating text at the same time as the card target's, in the same place, but respecting the AST's UI options (meaning, if the GNB has their 'buff numbers' turned off, and the AST has them on, the GNB won't see them, but the AST will).
Alternatively, the 'purple text' could go in the 'damage dealt' text stream, along with the Brutal Shell crit we're dealing to the enemy in this example. Might get a bit cluttered up there during burst windows, though, and obscure gameplay/hide boss model (so you can't see a model-based telegraph like 'it raised it's left hand for a slam' for example). Especially, for example, as a Monk, who would have GCDs, Autoattacks, extra Autoattacks from Riddle of Wind, and even more Autoattacks from a reworked Arrow
The AST's card effect is split from the rest of the buffs (div, litany, step) on purpose, because it's more like an attack than a standard buff. Perhaps options could be deep enough so that you could eg 'show only AST card effects, but not all other buffs'