I've been in contact over WhatsApp with Transavia and Schiphol, with varying degrees of success. I've pasted some of the conversation below as they may prove useful to others.

Transavia kept telling me KLM would give me the boarding pass for their flight. I repeatedly told them KLM had confirmed (over chat and by direct phone call) they would not, and asked how to check in with Transavia when in Amsterdam.

Finally we got to this:
"If KLM is assuring you that you cannot get your Transavia boarding pass in Leeds, in that case you will have to leave the airport in Amsterdam and do the check-in directly with us."

I asked "How can I find out if I can self check in with Transavia airside?"

The response... "With Transavia, if we are talking about the airport of Leeds you won't be able to since the flight is not departing from there. Gonçalo 🐧"

I responded "No, Amsterdam. Can I self check in with Transavia using the self check kiosks airside in Amsterdam? "

The response to this was "Just the fact that you do not have any information regarding the airside in Amsterdam already means that you won't be able to do self check-in. I'm sorry if this is a inconvenience. Gonçalo 🐧"

I was asked them for the information!!

The Schiphol adviser said:
"In the case that there is no transfer desk available, you can also simply go to the gate to get your boarding pass. Another option is to go to T3 and T9 for a self-service kiosk. At T3, T5 and T9 it is indicated above or on the counter what you can do. "

I asked "So even if I haven't checked in I can get my boarding pass at the gate, airside, without needing to go landside? "

Reply was "Yes. However, I do recommend first going to one of the desks I just listed, to try and get your boarding pass at the self-service kiosks."

I'd had a few mixed messages so asked "Also just above you (or a colleague) have said there is no transfer desk for Transavia. Does this mean that there is nowhere airside to check in for a Transavia flight or that there is no dedicated desk but airside check in is still possible at self service or at the gate? "

Response was "I completely understand that you want to make sure. However, as the check-in process is entirely under the airline's responsibility, it is a little difficult for us to guarantee anything. So, while it should be possible to get your boarding pass at the gate, it is still the airline who decides about this. And yes, it is true that there is no transfer desk for Transavia specifically (which is why we recommend going straight to the gate). The self-service kiosks are not for a specific airline, any passenger can use these. So what you said is indeed correct. "

The last response was a bit equivocal but I've got a bit more confidence that I can use a self transfer kiosk. The ones I've seen pictures of have "Air France KLM" written on them but the agent says the kiosks are for all airlines.

Do any of you very helpful people have any experience of checking in airside using the kiosks? Or getting a boarding pass at the gate?