ff games have more then those classic summons. They can even use ff10 summons anima, or even vanguard plus you got yojimbo, siran, carbuncle, fat chocobo/chocobo king. the ones from 12 tactics advance. there is a huge list of summonings SE can use. as for lore they could go the ff11 route and make it a once banned magic practice. Or go the ff9 route and make it a ancient practice. Or go the ff8 route and it be part of speciel military training. there are many ways it can fit with the theme.