It requires no such thing. it simply lists for forms of proof that are acceptable. The U.S. State Department will often accept a baptismal certificate in place of a valid birth certificate on visa applications. Is that a violation of the Constitution? If so, there are millions of foreigners in this country who need to be kicked out.
No, it's not a violation of the constitution because the constitution does not prohibit the government from using religious based records for visa applications. Furthermore, such people have options, and choose to use baptismal records.
However, the constitution explicitly forbids any religious test from being applied to a person in order to hold the office of President of the United States. The AZ law would seem to present such a test. The law refuses to honor certain types of birth records from other states. Candidates who come from states whose records are not honored by AZ would then be explicitly
required by the statute to present a baptismal record. This then creates the situation that all people from the many states who do not issue the restrictive class of records that AZ demands
must have been baptized. Any person who is from one of those states, and who was not baptized is deemed by the AZ law as ineligible to the Presidency. Therefore, the law is additionally unconstitutional because it
requires a religious test to be applied.