We live in Dhahran, Saudi Arabia. Can we apply or a china visa in Bahrain?

No, you cannot apply for China visa in Bahrain since you don't have a Bahrain work permit or resident permit. You need to submit the following requirements to related department (https://www.travelchinaguide.com/embassy/saudiarabia.htm ) in Saudi Arabia to obtain China tourist visa:
1)Original passport and a photocopy of its pages with personal information.
2)A fully completed Visa Application Form with the applicant's signature.
3)An original letter from the applicant's employer. It should contain the applicant's name, nationality, occupation, purpose of the visit, the planned date of entry and financial conditions.
4)A passport photo with white background.
5)If other additional documents are required, consular officers will contact the applicant directly according to the contact information left on the Visa Application Form by the applicant.