You would need to embed the logic within the procedure being called, not the procedure calling the procedure. So in the example you gave, each database's version of procedure...

If the stored procedures being called include metadata from INFORMATION_SCHEMA.ROUTINES in the RAISERROR, they should return the desired results.

CREATE PROCEDURE usp_ErrorTest

AS

DECLARE @ProcName nvarchar(128);

DECLARE @RoutineCatalog varchar(128)

DECLARE @RoutineSchema varchar(128)

SET @ProcName = OBJECT_NAME(@@PROCID);

select...