There was a time when Transavia checkin (at least for AMS departures) was actually handled by the KLM web checkin engine (you were discreetly "handed over" to KLM to use their check-in flow and ended up with what looked like a stripped-back KLM OLCI boarding pass); in the last few years, however, HV checkin appears to have been fully separated from its parent.

You always have to check in with the operating carrier, so yes, you have to check in with Transavia. Trying to check in with KLM (web, kiosk, agent) won't work.

Perhaps it's something along the lines of you're trying to check in at T-30h (as KLM allows), but Transavia only opens later (even in the days when you got routed via the KLM website, I frequently noted that checkin didn't open exactly on time as KLM does). I've not had much recent experience of checking in online with HV, but I do remember many instances in the past where OLCI wasn't open until quite a few hours after it was advertised as opening up.

So perhaps wait a few more hours, and then try again.

EDIT: That said, I have never booked (and therefore never check in for) a ticket where a Transavia flight connects with a KL flight on the same KL ticket. Perhaps, indeed, this restricts you to checking in at the airport with an agent at whichever HV outstation this is.

Please report back!