SET ROWCOUNT will affect the rows returned by both the outer SELECT statement and the outer level of any SELECT statements contained in the scalar UDF.
So this isn't possible through use of SET ROWCOUNT alone unless you can change the definition of the Scalar UDF.
I assume that you are using the undocumented and un guaranteed variable assignment approach. If you change to the XML PATH approach then as the top most SELECT only returns 1 row anyway it will remain unaffected by the SET ROWCOUNT 1 command.
Example Code
    USE tempdb;
GO
    CREATE FUNCTION dbo.Concat1(@number INT = 0)
    RETURNS NVARCHAR(max)
    AS
      BEGIN
          DECLARE @Result NVARCHAR(max)
          SELECT @Result = COALESCE(@Result + ',','') + name
          FROM   master..spt_values
          WHERE  number = @number
          RETURN @Result
      END
GO
    CREATE FUNCTION dbo.Concat2(@number INT = 0)
    RETURNS NVARCHAR(max)
    AS
      BEGIN
          RETURN
            (SELECT STUFF((SELECT ',' + name AS [text()]
                           FROM   master..spt_values
                           WHERE  number = @number
                           FOR XML PATH('')), 1, 1, ''))
      END
GO
    SET ROWCOUNT 1;
    SELECT dbo.Concat1(number) AS Concat1,
           dbo.Concat2(number) AS Concat2
    FROM   master..spt_values
    WHERE  number = 1
    DROP FUNCTION dbo.Concat1, dbo.Concat2 
Results
+---------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Concat1 |                                                                              Concat2                                                                               |
+---------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| rpc     | rpc,yes,autoclose,published,WINDOWS/NT,trusted,ignore duplicate keys,binary,varbinary,primary,NULL,Xact,NUL,GRANT,system table,disable_def_cnst_check,default disk |
+---------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------+