I would fully expect that you can get your Transavia BP in LBA. The flight you are on is a KL flight, coded with a KL flight number, and therefore its in KL's systems. This is no different from having a KL flight operated by Copa, an AF flight operated by Austrian, and so on.

At AMS the min connection time between KL and HV is 1:10. However this requires a through ticket. Picking up luggage is not part of that, and you will most likely not have time to do so. Again, I do not believe (at all) that this would be required - but if at LBA they can only check the bag till AMS please do know that KLM is responsible to get you to SVQ and pay out compensation. In case of an overnight hotel is to be provided for you too.

Getting a BP in AMS should be fine. This is a KLM ticket, so you should be able to use KLM's transfer machines or desk. Having to go landside won't work within the time, and I would suggest to not do so unless KL staff explicitly tells you to do so - as it likely will cause you to miss the flight.