At the same time, they could introduce (at least) two additional regular summons: one Ifrit-like, high damage summon with cast times that would focus on AoE (so somewhat of an Ifrit/Garuda mix), and an additional lower damage, instant cast summon, which would allow you to still gain three charges to summon Bahamut during combat phases that require heavy movement.

The Demi summons could also be changed such that they are somewhat of a choice of which to summon. Make it so that you can't summon the same Demi twice in a row within X minutes. One could have slightly higher single-target damage with no support effects, which would end up as the main boss summon, and the others could be more AoE/support focused, so that they would be more useful for dungeon trash fights, and provide more of a choice to use during boss fights for whatever support effects they provide.

Overall, this would give both a much greater 'Summoner' class fantasy, as nearly every actual effect would be coming from the summons themselves - pretty much the only spell that would be cast by the Summoner itself, other than commands, would be Aethercharge/Dreadwyrm Trance/Summon Phoenix/Solar Bahamut, Energy Drain/Siphon, and Resurrection (and, y'know, Physick, 'cause that's super important). Ruin/Fester/Painflare would all come from Carbuncle. All of the Gemshine/Precious Brilliance spells would be coming from the secondary summon

Nearly every summon would be able to stay on the field for at least a short duration, until their charges were used up, instead of just being a single attack when you summon them, so you'd actually feel more like you were summoning them. Ghosting would be less of a problem, since Carbuncle and the other summons are all no longer linked together. You'd have more interaction via weaving the summon commands, but as you would still mostly only be doing single weaves, it should still be viable for players on higher latency.

As many abilities would still utilize the same button changes as they currently do, adding a couple additional summons wouldn't create much button bloat for the job. Given the total number of summons available in the Final Fantasy universe, this would also open up ways to better add more summons for future level increases. For example, you could have the next expansion add variants for the Egi summons so that you would alternate between them for each rotation: Titan/Garuda/Ifrit on the first rotation, then Ramuh/Shiva/Leviathan the next, all of which are just slightly higher damage than their counterparts, then back to the first set.