No problems on the outbound. DL policy is to always thru check bags on separate tickets if the other carrier is KLM/AF/a couple of other partners.

The return will be trickier. Since it's doubtful you'll be checking in with actual KLM staff, I'd say your chances are increased. I find most contract staff (when not another airlines staff) will normally always thru check baggage. I'm not sure what KLM's official policy is. However, IME I've never had an issue thru checking bags when the checkin is with KLM and other flights are on DL.