Yet another thought popped into my head, for no reason: first, attach a good sized piece of plywood to the wall with several screws, using the spiral anchors to distribute the load above and below the load point.
Then, attach a piece of perforated steel angle to the plywood. The base of the bed will rest on the horizontal surface of the angle bracket.
This method, although not the prettiest thing in the world, will ensure that the load of the occupant in the bunk bed will be widely distributed, so that the "pounds per screw" will be minimized.
Also, and you probably already know this, the original bunk beds had a load limit of 140 pounds.
Finally, once you install the bunk, test it with big cinder blocks, piled one at a time onto the bunk. Weigh the blocks first. If you reach the weight of the intended occupant without structural failure, stop right there and declare victory!