I have a suggestion, based on what I think is an oversight. In the scripted_effects folder for Imperator: Rome, there is a file named 00_event_effects that contains add_relevant_subunit_effect and add_relevant_loyal_subunit_effect. These determine what types of cohorts are generated by events for different culture groups. Your mod doesn't have a version of this file, so for your new culture groups, the generated cohorts will fall into the else case, which includes chariots. I don't know if the game will actually allow you to get chariots this way when your military traditions don't allow chariots, but just in case, you should overwrite the default file to include your new culture groups.
