A simple solution is to change just the skin of the 3 basic primos. Keep the same power and effect, but change the skin only.

So with bahamut you summon the 3 original primos, with phoenix you summon the 3 new ones.

Ifrit > ramuh
Garuda > leviathan
Titan > Shiva

Garuda and leviathan are close in visual terms, tornado and wind vortex > tornado and water vortex.

Titan and shiva > replace the stone attack visual with ice

Ifrit and ramuh > change fire explosions to lightning