Legal Question in Education Law in California

No, you cannot block any of that information from the other school.

From the U.S. Dept. of Education:

Disclosure of Education Records

A school must:

Have a student's consent prior to the disclosure of education records;

Ensure that the consent is signed and dated and states the purpose of the disclosure.

A school MAY disclose education records without consent when:

The disclosure is to school officials who have been determined to have legitimate educational interests as set forth in the institution's annual notification of rights to students;

The student is seeking or intending to enroll in another school;

The disclosure is to state or local educational authorities auditing or enforcing Federal or State supported education programs or enforcing Federal laws which relate to those programs;

The disclosure is to the parents of a student who is a dependent for income tax purposes;

The disclosure is in connection with determining eligibility, amounts, and terms for financial aid or enforcing the terms and conditions of financial aid;

The disclosure is pursuant to a lawfully issued court order or subpoena; or

The information disclosed has been appropriately designated as directory information by the school.