On the topic of carbuncle, I am not bothered by him not attacking and can actually get behind the idea of him being a summoner's personal support pet; however, that comes with the caveat that he actually offers support (i.e. having more than just Radiant Aegis).

Ideally, what would really clean up the clunk of relying on carbuncle would be fixing whatever issue restricts a player from having more than 1 pet at a time. If carbuncle is not forcibly dismissed for each summon animation, it really opens up the door for him to be a genuine sidekick for you. This is 100% doable, it's just a matter of identifying the time required and making time for that task. Even if this would not come to fruition, I would focus on the following changes for the next expansion.

1. Return Searing Light to Carbuncle. If having more than 1 pet at a time cannot be fixed for whatever reason, I'd still make this change as it feels thematically appropriate since it originally belonged to carbuncle anyway. If absolutely needed, introduce a slightly weaker version of Searing Light (like a 2% buff instead of 3%) that replaces Searing Light if carbuncle is not in play. Optimally, you need to time it before summoning bahamut, but if you mess up, you aren't forced to delay the action till bahamut is dismissed.

2. Introduce a new action usable only when you have no gemstones to summon with (no ifrit, titan, or garuda). This action commands carbuncle to grant you a random gemstone and allows you to more effectively delay your next bahamut or phoenix summon. Additionally, you get another action that requires at least 1 gemstone be available. This action commands carbuncle to deliver a moderately potent attack by consuming all your gemstones. The potency increases the more gemstones you had. The potency is not worth the entire summon phase, but allows you to dump your gemstones if you were forced to hold onto a summon or two and need to go into your next bahamut/phoenix window.

3. Lastly, add a new action that harkens back to Alphinaud's moonstone carbuncle's field. This would be a cooldown centered around your carbuncle that increases HP restored to allies that stand within it; additionally, while you stand within it, the cast time of Resurrection is reduced to 0 seconds.

Something like this would make your carbuncle actually feel like a support pet to you and help you with survivability, recovery, and adding flexibility to your rotation.