Remove static pet. The goal for summoner should be to call forth your summon similar to how they do for Bahamut. So have it where the abilities are sequenced, and the final ability of the sequence determines what was summoned. It could be similar to ninja Mudra system in that regard, but it will always summon something and it's part of the cast rotation, not oGCD. Each summon has different effects and lasts for a short time.
For example Garuda could be the summon for AoE damage, so the final part of the sequence would use a wind spell. Ifrit could be for ST requiring a fire spell. This could even allow for a support type role too if they wanted to allow for short buffs or resource Regen from another summon.