All visitors to KSA can drive your insured vehicle within KSA and Bahrain, provided you have covee, and provided your insurance company policy states this.
Ace / Walla Insurance policy clearly states that anyone, without Iqama, can drive your vehicle with valid driving license from home country and if they are over age of 21.
Note this is for "visitor"; this is NOT applicable to dependents, etc, who have their own Iqama.
Anyone who has valid Iqama MUST driving on Saudi license.
The reason I state this is because for people who commute daily between KSA and Bahrain, your families in Bahrain can get visitor visas for KSA, and yes, effectively drive your vehicle in KSA, again with valid driving license from home country and if they are over the age of 21.
Conversely, all dependents in KSA can drive your vehicle in Bahrain, with valid driving license from home country and if they are over the age of 21.
They cannot cross the causeway and you don't need to inform insurance, be it verbally or in writing, as visitors have no Iqama records.