Usando a cláusula OUTPUT para inserir valor que não está em INSERTED

Use MERGE em vez de INSERT :

MERGE
INTO    trn_temp d
USING   (
        SELECT  D.DET_DATE, 'SOMETEXT' AS sometext, SUM(D.DET_NET) AS the_sum
        ...
        ) s
ON      (1 = 0)
WHEN NOT MATCHED THEN
INSERT  (TRN_TRAN_DATE, TRN_DESCRIPTION, TRN_AMT)
VALUES  (det_date, sometext, the_sum)
OUTPUT  s.*

Atualização:

Para contornar o GROUP BY problema, use isto:

DECLARE @tmp TABLE
        (
        det_primary INT NOT NULL PRIMARY KEY
        )

MERGE
INTO    register r
USING   detail d
ON      (r.det_primary_link = d.det_primary)
WHEN NOT MATCHED THEN
INSERT  (det_primary_link, ins_date)
VALUES  (det_primary, GETDATE())
OUTPUT  d.det_primary
INTO    @tmp;

INSERT
INTO    trn_temp (trn_tran_date, trn_description, trn_amt)
OUTPUT  INSERTED.*
SELECT  det_date, 'sometext', SUM(det_net)
FROM    @tmp t
JOIN    detail d
ON      d.det_primary = t.det_primary
GROUP BY
        det_date