Firstly, the power that the Engine can use is not "cast spells," it is that all spells in the chosen lore have a casting level of one less than normal, you still need a wizard who knows the spell to attempt to cast the spell.
Secondly, Yes you can choose a different lore at the start of each friendly magic phase, the only rule is that multiple EotG must choose different lores. So your EotG could choose Life one turn to help the Slann and Heavens the next to help the skink priest on the EotG.