A drywall or similar screw could fail, since they're not designed with shear strength (lateral load) in mind.
This is a job for lag screws. If you use a 3" lag screw into the studs, that will go through the frame, through 1/2" or 5/8" of drywall, and more than half the thickness of the stud. This doesn't look very deep, so you'll probably need to anchor it at each stud around the edge of the frame.
Alternately, you can use the newer "structural screws" which are more expensive, but require less pre-drilling. Either way, if anchored securely to each stud, the load limit of the fasteners will be higher than the load limit of the bed frame itself, since the cross-pieces are screwed in.
Here's a link to some more information about the differences and relative strengths.
And using a stout dresser as a support is wise, it will handle the outside load by compressing downward instead of shearing against the wall. This plan, if assembled correctly, should easily be able to support several hundred pounds.