If you are traveling without checked bags, do not go landside. LBA passengers will arrive at the terminal by bus at the beginning of the D-pier. Go upstairs and follow the signs to Schengen departures. You will now go through passport control, where, if needed, there is a short connection lane. Your HV fllight to Sevilla will leave from the C-pier or D-pier Schengen gates. If LBA is unable to give you a boarding pass, there is a big station for transfers in the Schengen lounge, on your left hand just before the split to (straight ahead) the C pier and (turning right) to the B-pier. Transavia is not handled by KLM anymore, but by Viggo.Check if there is a Viggo transfer counter there, if not you may ask KLM staff, but if you are really tight timewise head directly to the gate. If you do have checked bags, is KLM handling agent in LBA able to tag them through to Sevilla? They may be able to tag the bags through but not issue boarding passes, in that case do as above, get your boarding passes at the gate.
If you have checked bags and LBA can only tag them to Amsterdam, you will need to go landside: via passport control to baggage claim, then out, upstairs and to terminal 1 to check in with Transavia (check-in row 2/3). But this scenario is not going to work, as there will be waiting time to go through passport control and to wait for your luggage at the belt.
Are you traveling anytime soon? I may be able to check with Viggo at Schiphol how and what. Do print out a copy of your ticket in case passport control asks about it though.