The OP asked two questions.
Can you legally use reduced TOFF thrust?
(with no runway analysis (tabulated TOFF charts) available for the airport.)
And
Shouldn't the company have airport analysis for all the approved airports (Part C) to operate before even going there?
If the answer to the first is YES then the answer to the second should be NO.
I think that Left Coaster's "YES" was in answer the second question.