You are correct on the Sunni-Shi'ite conflict, but think Muslim Brotherhood as most of the populace in SA is not of a native tribe, but rather imported workers. Iran will over look sectarianism in the beginning to get the Caliphate established.
Most of the imported workers in SA, though, are Palestinians or Pakistanis -- thus Sunni.
Your assertion that Iranis would be willing to support a Sunni coup, though, is probably correct.