Have you tried requesting a Finnair e-ticket receipt? That will show the exact details of what is actually booked - fare basis code, fare booking code and so forth.

BAH can regard it as suspicious all they want that you take PDFs but I do the same (except as a screen recording or long screenshot) with any high value BA / BAH bookings. I want to have evidence of everything they told me that I'm relying on, e.g. double TP conditions, any bonus offers, plus the amount of Avios & TPs the calculator says I'll get.
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As a minimum, when booking into any premium cabin I take a screenshot of the cost in economy so that if I'm downgraded (e.g. due to cancellation and rebooking) I have a clear basis for establishing the fare difference which I'll ask for. On short haul flights this can be a lot more generous than the 30/50% Mennens entitlement.