I'm having a hard time following your comments. The specified PWHT temperatures are based on staying below the "lower transformation temperature". I don't remember anything in B31.3 that addresses the tempering temperature range.
To prove Code acceptability, a review of the MTR against ASME Section IIA is sufficient.
For ordering purposes, we state clearly on the PO: "The material must conform to (for example) SA335-P22; an MTR (traceable to the material supplied) must accompany the material delivery." We do not specify technical details, related to how to produce the product, to the supplier.
For ASTM materials: If they are used in an ASME Code "pressure boundary" appication, they must be acceptable (first) by the code of construction, and also be listed in QW-422 of Section IX. If either requirement is not met, the material is not used by us.
Charles