Error when using OPENROWSET from a View as a nonsysadmin
To respond to my own question, was able to get it working using a loopback linked server and using OPENQUERY instead of OPENROWSET. The non-sysadmin account now has the controlled access it needs.
Here is the new query the view uses:
SELECT *
FROM
OPENQUERY([LOOPBACKSERVERNAME], 
           'SET FMTONLY OFF; 
            SET NOCOUNT ON;
            EXEC [My Stored Procedure] @params = ''field1,field2''
            WITH RESULT SETS
            (
                (
                    field1 NVARCHAR(1000),
                    field2 NVARCHAR(1000)
                )
            )
        ')